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DEVELOPMENT LESSONS 

Caul 

IN 

1 W 5. 

MENSURATION 


WITH OVER A THOUSAND EXAMPLES AND PROBLEMS 
ILLUSTRATING THE PRACTICAL USE 
OF MENSURATION 


BY 

WM. F. SELL 

w 

Principal 21st District School, Milwaukee 


) > ))a 

—-nrn © > 

> > > ) 

> - ) > > > 


i > > > } > 3 


> > ) ) 9 


SECOND, EDITION, 

^ _ > 




a J 
o J o J 


A. FLANAGAN COMPANY, Publishers 


CHICAGO 




Q446T 

■ S 47 


LIBRARY of congress 
T wo Copies Received 


JAN 4 1904 

\ Copyright Entry 

c w 

CLASS <L XXc. No, 
■) *) 3 7- S 
COPY S 


Copyrighted, 1903, 
By 

Wm. F. Seil t 














PREFACE. 

This little book is not intended to be a scientific treatise of 
the subject of Mensuration, but is designed for pupils who cannot 
attend school long enough to study Geometry, and still ought to 
understand the most practical subject in Arithmetic. Nothing is 
attempted here that an average pupil in the upper grades cannot 
understand. 

In the subject of Mensuration, as well as in all things 
taught, one should start out right; start with correct and clear 
conceptions. To do this the teacher should begin with something 
concrete. In mensuration of surfaces, figures on the blackboard 
will do, but when dealing with solids the teacher should procure 
these forms in wood, or, what is sometimes better, cut them from 
an apple or a potato in the presence of the class. When pupils 
have a clear conception of the idea in the concrete they should 
be led to imagine, and later learn to reason in the abstract. 

All are agreed that a clear definition of terms is essential in 
the study of Geometry, but experience has taught many that the 
elements of many brandies can be taught without any attempt at 
exact definitions. We human beings, in fact, make very little at¬ 
tempt at defining. We know a man, a hat, a house, when we see 
it, without ever having learned to define it. In like manner will 
pupils know a square, an angle and a circle without knowing the 
definition, if the teacher applies these terms to certain figures and 
represented ideas. The author therefore suggests that the teacher 
of this subject apply correct terms to ideas correctly represented, 
and not stop to develop a definition. 

The chief aim of the teacher must be that each pupil has a 
clear conception of the step under consideration, before beginning 
the next, because in this subject especially does the next step al¬ 
ways depend upon a preceding one. 

WM. F. SELL. 

Milwaukee , Wis., Nov. 24, 1903. 


MENSURATION OF SURFACES. 


LESSON I. 

LINES. 

1. Draw a horizontal line 4 inches long. 

2. Draw a vertical line 6 inches long. 

3 . Draw two horizontal lines parallel to each other. 

4. Draw five vertical lines parallel to each other. 

5 . Draw a line parallel to the left edge of your paper. 

6. Draw a line parallel to the base of the blackboard. 

7 . Without measuring them draw lines of different lengths on 
slate, paper or blackboard. Estimate the length of each line. 
Test, and see how near you have come to the correct length. 

8. Estimate the length and width of your room, your desk, 
your school yard, the windows, the doors, the blackboard, the 
height of your room, the width of the road or street. Test each, 
and see how near you have come to it. 

9 . Name lines or edges that are parallel with the pieces of the 
window casings of your room. 

10. What lines or edges are parallel with the edges of your 
desk? 

11. Name other lines and edges in your room that are parallel 
to each other. 

12 . Draw a line perpendicular to another line, 

4 



LINES. 


5 



Note: In eadh of the above three cases c d is perpendicular to a b; that 
is, c d meets a b so that at their point of union a square corner, or a right 
angle, is formed. A vertical line is a straight up and down line. A line 
is perpendicular to another line when it meets that line at right angles. 

13. Draw a line perpendicular to the left edge of your paper. 

14. Hold your pencil perpendicular to the top of your desk; 
to the blackboard. 

15. Tell what lines, edges and plane surfaces are perpendicular 
to other lines, edges and plane surfaces in your room. 

16. Hold your book up in different positions and let another 
pupil hold his ruler perpendicular to the cover of the book. 


LESSON II. 

ANGLES. 

An angle is a corner, or a figure made by two lines which 
meet. 



a square corner, or angle. Such an angle is called a right angle. 









6 


MENSURATION. 


d e f forms a sharp, or an acute angle, ghi forms a blunt, or an 
obtuse angle. When an angle is larger than a right angle, it is 
an obtuse angle; when it is smaller than a right angle, it is an 
acute angle. 


A 



1. In the above figure, what kind of an angle is a o e? a o b? 
e o b? e o c? d o e? a o c? c o b? 

2. A circle is divided into 360 parts, called degrees. How 
many degrees in one half of a circle? In one quarter of a circle? 
In one eighth of a circle? 

3. a o b is an angle of how many degrees? 




ANGLES. 


7 


4. Of about how many degrees is the angle boc? doc? e o d? 
e o a? 

5. Draw an angle of about 60 degrees. Of about 135 degrees. 
Of about 100 degrees. 150 degrees. 30 degrees. 

6. Can you draw an angle of 180 degrees? Explain. 


LESSON III. 
RECTANGLES. 




The above three figures are rectangles because their angles are 
right or rectangles, a is a square; b and c are obkongs. 

1. What is the difference between a square and an oblong? 

2. Draw a square 4 inches each way. 

3. Draw a rectangle 6 inches long and 4 inches wide. 

A E> 


G 


D 






8 


MENSURATION. 


c d or a b is called the base ; a c or b d is called the altitude. 

4. In the above rectangle what can you say of the lines a b and 
c d as to their relation to each other ? 

5. What can you say as to their length? 

6. If a b is 10 inches, how long is c d? 

7. If a c is 4 inches, what is the length of b d? 

8. Draw a rectangle 6 by 8 inches. 

9. Draw one on the blackboard 2 by 3 feet. 

10. Point out five rectangles in your room and estimate the 
dimensions of each. 

11. Draw a 4 by 6 inch rectangle on your paper and letter it 
as the one above. 

12. Cut a piece of cardboard 1 inch square. This is called a 
square inch. 

13 . How many square inches can you lay along the line c d? 

14 . How many such rows can you lay in the rectangle ? 

15 . Then how many square inches in a rectangle 6 inches long 
and 4 inches wide? 

Note: If multiplication has been taught correctly, the only logical and 
correct way is to say, 6 times 1 sq. in. == 6 sq. in., (as there are 6 times one 
square inch along the line c d ) ; and, since there are four rows of 6 sq. in., 

4 times 6 sq. in. = 24 sq. in. But for all practical purposes it will be 
enough to say, the base times the altitude, or 6X4 = 24, calling it 24 square 
inches. 

16 . How many square inches in a rectangle 10 inches long and 

5 inches wide ? 

17 . What is the area of a rectangle 6 feet long and 3 feet wide? 

18 . The area of a rectangle 7 by 12 feet? 

19 . The area of a square 8 feet on a side ? One foot on a side ? 

20. Area of a rectangle 16 J by 12? 

LESSON IV. 

PROBLEMS IN RECTANGLES. 

1. Find the area of the top of your desk. 


PROBLEMS IN RECTANGLES. 


9 


2. The area of your school room. 

3. The area of the west wall of your room. 

4. The area of your blackboard. 

5. The area of the sidewalk in front of your house. 

G. The area of your mother’s sitting-room. 

7. Bring to class five other areas that you have found. 

8. Find the area of the figure below. 

9. Is there more than one way of finding the area? Try it. 
















10 


MENSURATION. 


10. What is the area of your school ground after deducting 
the area of the ground occupied by buildings ? 

11. What is the area of the four walls of your room after the 
area of the door and windows has been deducted? 

12. How much of the four walls of your room is not covered 
by blackboards ? 


LESSON V. 

LUMBER MEASURE. 

A board one foot long, one foot wide and one inch, or less, 
thick is called a board foot in lumber measure. If it is thinner 
than one inch it is figured the same as one inch in thickness; if 
thicker than an inch it is figured exactly according to its thickness. 
A board 12 feet long, one foot wide and 1J inches thick contains 
15 feet of lumber; 1 J inches thick, 18 feet; inch thick, 12 feet. 
Usually, however, dealers do not figure a fraction of an inch in 
thickness, but set the price in proportion, and sell according to 
surface only, and in odd widths and thicknesses according to 
linear feet. 

1. How many feet of lumber in a board 16 feet long, 1 foot 
wide and 1 inch thick? 2 inches thick? 1J inches thick? 1J 
inches thick? If inches thick? inch thick? 

2. Find the number of feet of lumber in a board 18 feet long, 
1 inch thick and 10 inches wide. 14 inches wide. 8 inches wide. 

3 . How many feet of lumber in the same boards 2 inches thick? 
| inch thick? 1J inches thick? 2^ inches thick. 

4 . How many feet of lumber in a scantling 12 feet long, 4 
inches wide and 2 inches thick? In a joist 16 feet long, 6 inches 
wide and 2 inches thick ? 

5 . Find the number of feet of lumber in the following twenty 
examples: 


LUMBER MEASURE. 


11 


LENGTH. 

WIDTH. 

THICKNESS. 


LENGTH. 

WIDTH. 

THICKNESS. 

8 ft. 

10 in. 

1 K in. 


14 ft. 

4 in. 

X 

in. 

12 ft. 

6 in. 

2 in. 


10 ft. 

8 in. 

6 

in. 

10 ft. 

14 in. 

2 in. 


18 ft. 

10 in. 

2^ 

in. 

16 ft. 

8 in. 

2 y 2 in. 


16 ft. 

14 in. 

2 

in. 

14 ft. 

12 in. 

3 in. 


12 ft. 

6 in. 

2 

in. 

18 ft. 

6 in. 

4 in. 


14 ft. 

8 in. 

8 

in. 

8 ft. 

4 in. 

1-U in. 


8 ft. 

4 in. 

4 

in. 

12 ft. 

11 in. 

1% in. 


6 ft. 

14 in. 

8 

in. 

16 ft. 

9 in. 

2 in. 


8 ft. 

10 in. 

10 

in. 

14 ft. 

5 in. 

3 in. 


20 ft. 

12 in. 

12 

in. 


6. If an inch board 1 foot wide and 16 feet long is laid on 
the floor, how much surface will it cover ? 

7. How much surface will 10 such boards cover? 

8. How much surface will these boards cover if 2 inches thick? 
If 1-J inches thick? Has thickness anything to do with the cov¬ 
ering of surfaces? 


LESSON VI. 

KINDS OF LUMBER. 

Scantlings are usually 3 or 4 inches wide, and 2, 3 or 4 inches 
thick, and are used upright in the frame of a building. A 4 x 4 
scantling is 4 inches wide and 4 inches thick, and is used in the 
corner of the frame. A 2 x 4 scantling is used between the cor¬ 
ners. Scantlings used as above are called studding. 

Joists are usually 2 x 6, or 2 x 8, and are set on edge to sup¬ 
port floors. 

Timbers are usually 6 x 6, or larger, and are laid on the founda¬ 
tion of buildings, where they are called sills ; when used in the 
upper part of buildings they are called beams. 

Sidings are usually J inch thick and 6 inches wide, and are 
laid one inch over each other on the outside of frame houses. 

Planks are usually H to 3 inches thick, and are used for side¬ 
walks, heavy floors in barns, and bridges. 














12 


MENSURATION. 


Flooring is usually 1 , 1J or 2 inches thick and 2 to 6 inches 
wide, and is used for floors and wainscoting. 

Fence boards are usually 6 inches wide and 1 inch thick. 

Rafters are scantlings used to support a roof. 

Stringers are scantlings used to support a sidewalk, or for 
forming the horizontal parts of a tight board fence. 

Laths are sold in bunches of 50 , and a bunch covers 3 square 
yards. 

Shingles are sold in bunches of a quarter thousand; 10 shingles 
laid 4 inches to the weather will cover one square foot. 

1. How many feet of inch lumber 1 foot wide and 16 feet long 
will it take to cover a floor 16 feet long and 12 feet wide? How 
many feet of lumber will it take if boards are only 6 inches wide ? 
If boards are 12 feet long and 12 inches wide? 

Note: Lumber is usually cut 10, 12, 14, 16 or 18 feet in length, hence, 
in practical calculations, an allowance must be made for waste in covering 
surfaces of odd dimensions. 

2. How many feet of inch lumber will it take to cover a wall 
20 feet long and 10 feet high? To cover a wall 24 by 10 feet? 

3 . A building is 20 feet long, 16 feet wide and 10 feet high; 
find the dimensions of one side wall. Of one end wall. 

4 . How many walls are there? How many square feet in 
each? 

5 . How many feet of inch lumber will it take to cover the 
four walls of a building 15 by 30 feet and 9 feet high ? How many 
feet will it take to cover the floor? The ceiling? 

6. The best way to find the dimensions of the four walls of a 
room or building is to multiply the distance around the room or 
building by its height. Explain why this will give the same 
answer as the one obtained by finding the dimensions of each wall 
separately. 

7 . Measure your mother’s sitting-room, and determine the 
amount of inch lumber necessary to cover wall, ceiling and floor. 


KINDS OF LUMBER. 


13 


8. If the roof of a building is 22 feet long and the rafters are 
12 feet long, how much inch lumber will it take to cover one side 
of the roof? Both sides? 

9. How many bunches of shingles will it take to cover that 
roof? 

10. Find the number of bunches of laths it will take to cover 
the four walls of the building 18 x 20, and 10 feet high. 

11. If the floor of this building is covered with 1J inch floor¬ 
ing, how much lumber will it take? If 2 inch plank is taken? 

12. Lumber is sold by the thousand feet, written per M. If 
lumber is sold at $20 per M, find the cost of 1,500 feet. 

13. Find the cost of 200 feet Of 18 feet. 

14. If the lumber in example eight costs $18 per M, find cost? 

15. If scantling cost $21 per M, what will fifteen 4x4 scant¬ 
lings, 12 feet long, cost? 

16. Find the cost of twenty joists, each 2x6, 12 feet long, at 
$20 per M. 

17. What will the shingles in example nine cost at $3.50 per 
M ? Remember that parts of bunches cannot be bought. 

18. Find the cost of this bill of lumber: 

12 2x4 scantlings, each 16 feet, at $16.00 per M. 

10 2 x 6 joists, each 14 feet, at $18.00 per M. 

5 2 x 12 planks, each 16 feet, at $20.00 per M. 

3 8x8 timbers, each 20 feet, at $25.00 per M. 

LESSON VII. 

PROBLEMS IN LUMBER. 

1. A barn is 40 x 80 feet and 20 feet high to the eaves. The 
rafters are 29 feet long. Cover it and the roof with inch boards 
costing $18 per M. It takes 800 feet to cover the two gable ends. 
Shingles cost $3.25 per M. A floor, 20 feet wide, is laid across the 
barn. The floor is made of 2 inch planks costing $20 per M. Find 
the cost of all. (Note that the floor is as long as the barn is wide.) 
A fraction of a board foot is figured as a whole foot. 


14 


MENSURATION. 


2 . A sidewalk is built of 1 £ inch lumber costing $22 per M. 
The boards are nailed on two 4x4 stringers costing $18 per M. 
What is the cost of a walk 10 rods long and 6 feet wide ? 

3. Find the cost of building a solid board fence 5 feet high 
and 120 feet long. The boards cost $20 per M and the two 2x4 
stringers upon which the boards are nailed cost the same per M. 
The posts cost 20 cents apiece, and they are set 8 feet apart. 

4. What is the length of the boards you are going to buy for 
this fence ? What length of scantlings will you buy for stringers ? 
Think carefully before you decide upon the number of posts. 
What length of boards will you buy for the walk in example two ? 

5. An open board fence, made of 6 inch fence boards, 6 boards 
high, is built around a lot 70 x 140 feet. The posts are set 7 feet 
apart and cost 22 cents apiece. The lumber costs $18 per M. 
Find cost. What length fence boards will you buy? Why are 
stringers not necessary here? Never carry any more than three 
decimal places. A fraction of a cent in the total is called a cent. 

6 . Build a walk, 6 feet wide, around the outside of this lot, 
using lumber and prices as given in example two. Find cost. 

7. Build a similar walk around the inside of this lot and find 
cost. 

8 . Find the cost of building a shed 20 x 40 feet and 10 feet 
high to the eaves. Let all lumber used in it cost $20 per M and 
shingles $3 per M. As studding, use 4 x 4’s on each corner, and 
2 x 4’s every two feet between. Rafters are of 2 x 4’s, 15 feet 
long, and two feet apart. Use 6 x 6 timbers as sills, and as plates 
for the rafters to rest on, nail two 2 x 4’s on top of each other. 
It takes as much board and 2x4 scantling for the two gable ends 
as it does for one end of the shed. 2 x 8 joists two feet apart 
are used to support the floor. The floor is made of 2 inch plank. 

9. What length scantlings are you going to buy for the raft¬ 
ers? For the studding? What length joists should you buy? 
Figure carefully the number of scantlings you need for studding. 
Number of joists. Number of rafters. 


PROBLEMS IN SURFACE MEASURE. 


15 


LESSON VIII. 

PROBLEMS IN SURFACE MEASURE. 

1. A school room 24 x 36 feet and 15 feet high, has eight 
3x6 foot windows in it, each 3 feet from the floor, and two 4x7 
doors. It has 3 feet of wainscoting all around and a 3 foot wide 
slate blackboard all around. Slate board is set over the plaster; 
wainscoting takes the place of plaster. The ceiling is of wood. 
At 12 cents a square yard what will it cost to paint the ceiling 
and wainscoting? 

2. At 25 cents a square yard what will it cost to plaster the 
walls ? Take out door and window space. 

3. What will the slate blackboard cost at 50 cents a square 

foot? 

4. At 22 cents a bunch what would the laths for the plaster 
cost? 

5. Work the same examples, using the dimensions and ar¬ 
rangements of your own school room. 

6. A sitting room 15 x 17 feet and 12 feet high has two doors 
and three windows, same size as those in example one. It has a 
one foot base board. At 24 cents a square yard what will it cost 
to plaster walls and ceiling? 

7. What will the laths cost at 27 cents a bunch? 

8. Carpet it with Ingrain at 60 cents a yard. Find cost. 

Note: Carpets usually look best when laid lengthwise. Ingrains are 

a yard wide. Brussels are usually 27 inches wide. Find out from the 
width of the room, how many strips of the carpet’s width are necessary 
to cover the floor, remembering when only a fraction of a strip is needed 
a whole strip of the room’s length must be bought. 

9. Carpet the room with Brussels at $1.20 a yard. Find cost. 


16 


MENSURATION. 


10. Wall paper is 1J feet wide and is sold in rolls of 24 feet, 
or usually in double rolls of 48 feet. Paper hangers usually de¬ 
duct 3 feet in width for each door and window. If a wall is 18 
feet long, how many strips will it take to cover it? 

11 . How many feet of wall paper will it take to cover the walls 
of a room 12x15 and 10 feet high? How many double rolls? 
How many double rolls if two doors and three windows are taken 
out? 

12. If wall paper costs 20 cents a double roll, what will it 
cost to paper the room in example six ? 

13. At that price, what would it cost to paper the school room 
in example one? 

14. At 25 cents a double roll, what will it cost to paper your 
mother's sitting-room? 

15. What will Ingrain carpet cost for mother’s sitting-room at 
60 cents a yard? 

16. What would a Brussels carpet cost at $1.10 a yard? 


LESSON IX. 

MORE PRACTICAL PROBLEMS. 


17 


F 

>ORCH 

WINDOW 2X3 

Closet 

4X6 

DOOR 

Pantry 

4X10 

DOOR 


WINDOW 


K(TCHtrJ 


9X16 


2 

On 

* 


£ 

§5 

r 


* 

§x 
- m 
$ 


WINDOW 


5lTTIN0 f^OOM 


20X24 


Bath F^oom 


axi 2 


DOOR 


Bed Room’ 

11X12 


t Z 

*o 

$ 


n 

& 

o 

Cl 


DOOR 


Reception Hall 


12 X 15 


DOOR 4X9 


O 

O 

o 

n 

o» 

x 

<0 


DOOR 5X9 


Parlor. 

15 X 16 


WINDOW 5X7 


Porch 


4Z 

y o 
'Jo 
$ 















18 


MENSURATION. 


This is the floor plan of a plain, rectangular house 32 x 48 on ; 
north-west corner lot 40 x 120 feet. It stands 20 feet from the 
sidewalk in front, and one foot from the north lot line. The room' 
are 12 feet high and reach to the eaves. It has a plain roof, reach¬ 
ing over the porches. 

Doors are 4x9, and windows 4x7, unless otherwise indicated. 

1 . What is the cost of a tight board fence on the north and 
west side of the lot? Fence 5 feet high; posts 10 feet apart; two 
2x4 stringers; lumber in the fence costing $20 per M; posts 18 
cents apiece. 

2 . What will it cost to sod the yard at 12 cents a square yard? 

3. What will a 6 foot sidewalk of 1 J inch lumber cost on the 
south and east side ? Use three 4x4 stringers. Price of lumber 
$25 per M. 

4. What would the same sidewalk cost when made of cement 
costing 90 cents a square yard? 

5. Cost of a 3 foot sidewalk from the front walk to the back 
of the house. Use inch lumber and two 2x4 stringers. Lum¬ 
ber, $24 per M. 

6 . Cost of 10 x 10 timbers for sills, and one beam running 
lengthwise, at $27. 

7. Cost of 2 x 10 joists, running from center beam to sides, 
two feet apart, costing $ 21 . What length of joist would you 
buy? 

8. Cost of the 4 x 4’s used on every corner of outside and par¬ 
titions, at $ 20 . 00 . 

9. Cost of the 2 x 6’s used as rafters, set two feet apart, each 22 
feet long. Price, $18.00. 

10. Cost of the roof boards, extending a foot over each gable 
end. Price, $16.00. 

11 . Cost of the shingles at $3.50 per M. 

12. Find the cost of the floor in the porches at $25.00. 

13. Hard wood flooring costs $32 per M. It is 3 inches wide, 
but on account of the matching it lays only 2 | inches. What will 


MORE PRACTICAL PROBLEMS. 


19 


the floor in the reception hall cost? In the sitting room? Bath 
room? 

14. At 18 cents a square yard what will it cost to plaster the 
sitting-room, deducting doors, windows and a base board one 
foot wide? 

15. What will it cost to plaster the parlor? 

16. At 20 cents a double roll find what it will cost to paper the 
bedroom. The sitting-room. (Include ceiling in both.) 

17. There is a 5 foot wainscoting in the bath room and the win¬ 
dow is 4 feet above the floor. Cost of wainscoting at $28 per M. 
What will it cost to plaster the bath room? 

18. What will a Brussels carpet, costing $1.50 a yard, cost 
for the parlor ? 

19. An Ingrain at 50 cents a yard for the bedroom will cost 
how much? 

20. At 12 cents a square yard, what will it cost to paint recep¬ 
tion hall, bathroom and kitchen, deducting door and window 
openings? 

21. At 22 cents a bunch, what will the laths for the closet and 

pantry cost ? 

22. At prices given in these problems, construct other prob¬ 
lems from the accompanying diagram. 

LESSON X. 

RHOMBOIDS AND TRIANGLES. 


A 



This figure, abed is called a rhomboid, ab — cd and they 
are parallel, a e is the altitude, it being the perpendicular distance 
between the two parallel sides of the rhomboid. 







20 


MENSURATION. 


1. Cut a similar rhomboid from card board, and cut off the 
triangle ace. Fit this triangle on the side b d. What kind of a 
figure have you now ? What is its area ? What is the area of your 
rhomboid ? How, then, do you find the area of a rhomboid ? 

2. If the base of a rhomboid is 12, and the altitude 7, what 
is its area? 

3. If the altitude is 16 inches, and the parallel sides 24 inches, 
what is the area? Parallel sides 18 feet, and altitude 10 inches? 

4. Cut a rhomboid 5 inches long and 3 inches wide from a 
piece of paper. Prove that its area is 15 square inches. 

A Triangle is a figure having three angles and three sides. A 
right triangle has one right angle. 

A B 



5. If a b is 6, and a c is 3, what is the area of this rectangle? 

6. What is the area of half of it ? 

7. Can you see that the dotted line c b divides this rectangle 
into two equal parts, and that the triangle cab, or the triangle 
c d b, is one-half of the rectangle abed ? 

8. What is the area of the triangle cab? Triangle c b d? 

9. In the upper triangle a b is the base, and a c or b d the alti¬ 
tude. In the lower triangle c d is the base and b d or a c the alti¬ 
tude. In the lower triangle c d is the base, and b d or a c the alti- 

10. What is its area? 



RHOMBOIDS AND TRIANGLES. 


21 


11. What is the base and altitude of the upper triangle? 

12. If this upper triangle is one-half of the rectangle, what 
is its area? 

13. If the area of a rectangle equals the base times the alti¬ 
tude (B x A), and a triangle is one-half of a rectangle of the same 
dimensions, how, then, is the area of a triangle found ? 

Area of Rectangle = B X A. 

Area of Triangle = £B x A. 

A 








22 


MENSURATION. 


15. The triangle a b c is an isosceles triangle, the side a b= 
the side a c. Name the altitude of this triangle by letters. Its 
base. If the base is 8, and the altitude 5, what is the area of the 
triangle a b c? The addition of the dotted lines a d and d c makes 
this into what kind of a figure? Is a c d an isosceles triangle also? 
What are its dimensions? What is its area as compared with the 
area of ab c? 


B 



16. The triangle a b c is an equilateral triangle, the three sides 
being equal. Name its altitude. Its base. If its base is 6, and the 
altitude 5, what is the area of the triangle ab c? The addition of 
the dotted lines b d and c d makes this into what kind of a figure ? 
Is b c d an equilateral triangle also ? What are its dimensions ? 
Its area ? Is an equilateral triangle an isosceles triangle also ? 



RHOMBOIDS AND TRIANGLES. 


23 



17. The triangle ab c is a scalene triangle. You will have no¬ 
ticed by this time that the altitude of any triangle is the perpen¬ 
dicular distance from the base to the top, or apex. Name the 
altitude of the triangle ab c. The base. If the base is 16, and the 
altitude 10, what is the area of a b c ? What is the area of a b d? 
Can you see that every triangle can be extended into either a 
rectangle, as in the case of a right triangle, or a rhombus, as in the 
case of the equilateral triangle, or into rhomboids as in the case 
of the isosceles or scalene triangles? What can you say of the 
dimensions of each triangle and its corresponding four-sided 
figure? 

18. The base of a triangle is 8 feet, the altitude is 12 feet; find 
its area. 

19. Stake off a large triangle on the school ground, and find 
the area of it by measuring its base and altitude. 

20. If a barn is 40 feet wide and the perpendicular distance 
between the eaves and the ridge pole is 20 feet, how much inch 
lumber is necessary to cover the gable end ? To cover both ends ? 





24 


MENSURATION. 


LESSON XI. 

TRAPEZOIDS AND POLYGONS. 



1. A necessary condition in a trapezoid is that the two bases 
are parallel. In the above trapezoid a & is one base and c d an¬ 
other. Can you see that the dotted line c b divides the above 
trapezoid into two triangles? Name them. 

2. What dimensions of the triangle c a b do you wish to know 
in order that you may find its area? 

3. If a b is 8, and a c 4, find the area of that triangle. 

4. If the bases are parallel, what is the altitude of the triangle 
c b d ? If c d is 12, what is the area of c b d ? 

5. What is the area of the whole trapezoid ? 

6. Draw a trapezoid on the board with one base of 20 inches, 
the other base of 30 inches, and the altitude 10 inches. Divide 
it into triangles and find its area. 





TRAPEZOIDS AND POLYGONS. 


25 



7. Draw a dotted line from c to b. What two triangles have 
you? 

8. If c d is 18, a b 12, and the altitude 6, find the area of a b c d. 

9. Find the area of a trapezoid with one base of 300 feet, an¬ 
other base of 640 feet, and an altitude of 120 feet. 

10. Draw a large trapezoid on a piece of paper. Find its 
dimensions, and its area. 

11. Draw another trapezoid on the blackboard. Measure it 
and find its area. 

12. Stake off a piece of ground in the shape of a trapezoid and 
find its area. 

In finding the area of a trapezoid the quickest way is to mul¬ 
tiply the sum of the bases by one-half the altitude. 

Area of Trapezoid = Sum of Bases x £ A . 

We can find the area of any irregular figure by dividing it into 
triangles, and finding the area of the triangles of which it is com¬ 
posed. 




26 


A 


MENSURATION. 



D 

13. The problem is to find the area of the irregular polygon 
abed. If c b is 8 and d e 6, what is the area of the triangle cb d? 

14. If a b is 7, and a c 3, what is the area of the triangle ab c? 

15. What is the area of the polygon ab d c? 



from d to e = 5 \ f g = 6 ; perpendicular distance between a b and 
c d — 4; between d e and f g = 5. 









TRAPEZOIDS AND POLYGONS. 


27 



17. Find the area of the above polygon. From / to b = 16; 
f to c = 20; £ to c = 18; the perpendicular distance from the line 
f b to a = 5; f c to b = 10; f c to e = 8; e c to d — l. 

18. Draw a similar polygon on the blackboard and find its 
area by measurements. 

19. Stake off an irregular piece of ground on the school ground 
and find its area. 





28 MENSURATION. 











THE HYPOTENUSE. 


29 


side, as is done in the figure above. Determine the center of 
the square on a b by drawing its diagonals. Through this center 
draw two lines at right angles to each other, and parallel to the 
sides of the square constructed upon the hypotenuse, b c, as is 
shown by the dotted lines in the above figure. This divides the 
square on a b into four parts. Cut out these parts, and also the 
square on a c, and place the five pieces so as to cover the square 
on the hypotenuse. (Place the small square diagonally in the 
middle, and the other four pieces will fit exactly around it.) What 
do you conclude? 

1. Try this again with a right triangle whose base is 4 inches 
and altitude 3 inches. It works every time. How many square 
inches in the square constructed on the base? In the square con¬ 
structed on the altitude? In both together? How many square 
inches must there be, then, in the square constructed on the 
hypotenuse? If there are 25 square inches in the square on the 
hypotenuse, what is the length of the hypotenuse of a right 
triangle whose dimensions are 3x4 inches ? How did you get it ? 

2. Try the same with a triangle whose base is 8 and alti¬ 
tude is 6. 

The square of the base + the square of the altitude = the 
square of the hypotenuse. 

B 2 + A 2 = h\ or 
H=VB , + A‘, or 

The hypotenuse = the square root of the square of the base 
+ the square of the altitude. 

The square of 8= 64 

The square of 6 = _36 
100 

The square root of 100 = 10, the length of the hypotenuse. 

3. Base is 16, altitude 12, find the hypotenuse. 

4. Base is 80, altitude 60, find the hypotenuse. 

5. Base = 172, altitude = 129, what is the hypotenuse? 

6. If 9 + 16 = 25, 9 — 25—what? If a + b = c what is a? 


30 


MENSURATION. 


7. If the square of 3 + the square of 4 = the square of 5, what 
does the square of 4 equal ? 

8. If the square of the base + the square of the altitude — the 
square of the hypotenuse, what does the square of the base 
equal ? 

9. When you have the square of the hypotenuse given, and 
the square of the base, what would the square of the altitude 
equal ? 

10. If the square of the hypotenuse is 25, and the square of 
the base is 16, what is the square of the altitude? What is the 
altitude ? 

11. If the base is 4, and the hypotenuse 5, what is the alti¬ 
tude ? How did you get it ? 

12. You have learned that the square of the hypotenuse, 
minus the square of one side, equals the square of the other side, 
hence, the other side equals the square root of this difference. 
Hypotenuse 10, altitude 6, find base. 

13. Find the side indicated by a dash: 


BASE. 

ALTITUDE. 

HYPOT. 

BASE. 

ALTITUDE. 

HYPOT. 

a. 

_ 

30 

34 

k. 

_ 

60 

75 

b. 

10 

24 

— 

1. 

40 

— 

50 

c. 

7 

— 

25 

m. 

X 

% 

— 

d. 

— 

80 

82 

n. 

18 

24 

— 

e. 

20 

— 

29 

0. 

— 

48 

52 

f. 

12 

16 

— 

P- 

21 

72 

— 

g- 

— 

40 

41 

q- 

36 

— 

60 

h. 

9 

— 

15 

r. 

90 

— 

150 

i. 

— 

15 

17 

s. 

5 

— 

13 

j- 

14 

— 

50 

t. 

36 

105 



14. A tree broke off, and the top fell 36 feet from the stump; 
the part that broke off was 60 feet long, how long was the 
stump ? 

15. How far is it from one corner to the opposite corner of 
your school room? 













THE HYPOTENUSE. 


31 



40 


16. If the top, or ridge pole, of a roof is one fourth of the width 
of a building above the horizontal line connecting the two eaves, 
the roof is said to be a quarter pitch ; if one half the width of 
the building, a half pitch. From this we may determine the length 
of the rafters. What is the length of the rafters for a quarter 
pitched roof in a 40 foot wide building? 

17. For a half pitched roof? 

18. A building is 20 feet wide; what is the length of the rafters 
in a half pitched roof ? 

Note : A practical carpenter, however, will not stop to figure the length 
of rafters. He will nail together a right triangle made of lumber, and 
measure the length of its hypotenuse, which is the length of the rafters. 







32 MENSURATION. 

LESSON XIII. 

PRACTICAL PROBLEMS. 

1. The base of a rectangle is 4, the altitude is 3, what is the 
area? 

2. The area of a rectangle is 12, the base is 4, what is the 
altitude? How did you get it? 

3. The area of a rectangle is 20, the base is 5, what is the 
altitude ? (5 x what = 20 ?) 

4. Find the base of a rectangle whose area is 640, and the 
altitude is 32. Area 84, base 12, find altitude. 

5. Draw a 4 x 6 inch rectangle. Draw a diagonal of that rect¬ 
angle, a line from one corner to the opposite corner. You now 
have two equal triangles. What is the area of one triangle? 

6. The base of that triangle is 6, what is the altitude? 

7. The area of a triangle is 20, the base 5, what is the altitude ? 
How did you get it? 

8. The area of a triangle is 40, the base is 10, what is the 
altitude ? 

9. Area 320, base 64, find altitude. 

10. What is the length of a farm 80 rods wide, and containing 
75 acres? 

11. The area of a field is 60 acres, one side is 60 rods, what will 
it cost to build a fence around it at $1.20 per rod? 

12. The area of a triangular field is 60 square rods, one side is 
30 rods, what is the distance around it? 

13. The longest side of a right triangle is 50 yards, and 
the shortest side is 90 feet, what is the length of the third side ? 

14. What is the distance around a right triangle whose base 
is 112 feet, and altitude 28 yards? 

15. What is the distance around a rectangle whose base is 100, 
and whose diagonal is 125 ? 

16. If a pole 96 feet high casts a shadow 128 feet long, what 
is the distance from the top of the pole to the farther end of the 
shadow ? 

17. A six acre rectangular field is 660 feet long; how far 
around it? 

18. If the foot of a ladder 50 feet long is put 30 feet from the 
base, how far up will it reach? 


PRACTICAL PROBLEMS. 


33 


DOOR 


WINDOW 


o H 

- s 

Z 


000 Q 


* < 
X Z 

o H 

5 


C? 

z 

o 


CP 
rr 
5 0 

o 

2 


*> 

err 


OOOR 


DOOR 


e/> 

3 

5 Z 
x O 

o 

o 


WINDOW 


WINDOW 









34 


MENSURATION. 


This diagram is the floor plan of an ordinary frame house, 40 
feet Jong, 24 feet wide and 13 feet high to the eaves. The win¬ 
dows, 3x5 feet, and the doors, 3x7 feet, are indicated. The 
house faces the south. 

19. How many square feet of ground does the house cover? 

20. If it stands in the middle of a square lot containing 2,500 
square feet, how much of the lot is not covered by the house? 

21. How many square feet in the lot lying north of the house? 

22. Lying w r est of the house? 

23. What is the distance around the house? 

24. Distance around the lot ? 

25. How far is it from one corner of the lot diagonally across 
to the other corner? 

26. How far is it from one corner of the house to the same 
corner of the lot? 

27. How far is it from the north-east corner of the house due 
east to the lot line ? 

28. Due north to the lot line? 

29. How far is it diagonally across the house? 

30. From the south-west corner of the house to the north¬ 
east corner of the lot ? 

31. If posts are set 10 feet apart, how many posts are needed 
around the lot? 

32. Put up a tight board fence, 6 feet high, on three sides of 
the lot. Use 2x4 stringers to nail boards on. If the lumber 
costs $25 per M, what would the lumber of that fence cost? 

33. Put up a picket fence in front of the house, and use 
stringers as above. How many feet of lumber in the stringers? 
How many pickets, each 4 inches wide, and set 4 inches apart, 
are needed? 

34. What length of boards should you buy in example 32? 
What length scantlings ? 

35. At 25 cents a square yard, what will it cost to paint the 
board fence ? 


PRACTICAL PROBLEMS. 


35 


36. Build an 8 foot walk of 1J inch lumber, costing $25 per 
M, on the street in front of the house, on three 4x4 stringers, 
costing $20 per M. Find cost. 

37. Find cost of a similar sidewalk from the front of the 
house, flush with the west side of the house, to the street. 

38. Build a 4 foot walk on the east and north sides of the 
house, and on the south side to the front walk. Use inch lumber, 
and only two stringers. Find the cost of the lumber at $25 per M. 

39. The sills under the house are 12 x 12 timbers, how many 
feet of lumber in them ? 

40. Three pieces of timber, 8x8, are laid across the width of 
the house, how many feet of lumber in them? 

41. Place these three pieces of timber in such a manner that 
the foundation rectangle is divided into four equal rectangles. 
Without considering the width of the timbers, what would be the 
width of each rectangle ? 

42. Place 2x8 joists across the width of these rectangles, each 
2 feet apart, and find the number of feet of lumber in the joists. 

43. Remember, the house is 12 feet high to the eaves from the 
top of the sill; place 4x4 scantlings at each joint where two 
walls meet. Find how many feet of lumber in the scantlings. 

44. If 2x4 scantlings are placed every two feet, what is the 
cost of the scantlings around the outside wall at $22 per M ? (Do 
not count the 4x4’s already up. Doors and windows not con¬ 
sidered.) 

45. In like manner find the cost of the scantlings in the par¬ 
titions. 

46. All the scantlings in the walls and partitions are up. Now 
plates are necessary to hold them together at the top. Two 2 x 4’s 
are usually nailed one on top of the other. What would they 
cost at $22 per M ? 

47. Joists, 2x8, are necessary to support the ceiling. They 
are nailed to each scantling across the building, and are two feet 
apart. At $24 per M what would they cost ? 


36 


MENSURATION. 


48. Put on the rafters that are to hold the roof. If they are 
placed two feet apart, how many does it take for one side of the 
roof ? How many for both sides ? 

49. The ridge pole, or highest point where the two sides of 
the roof meet, is 12 feet high from the top of the end scantlings 
already up; how long must the rafters be if they are extended 
one foot over the eaves? 

50. If the rafters are 2 x 4 scantlings, how many feet of lumber 
in all of them? 

51. How many feet of lumber in the scantlings needed for the 
gable ends ? 

52. Now the frame work is up. Cover the four walls with clap¬ 
boards at $15 per M. Find how much they will cost. (Doors and 
windows are not taken into consideration.) 

53. What would it cost to cover the gable ends ? 

54. Find the cost of the boards that cover the roof, extending 
them one foot over each gable; price, $16 per M. 

55. At 90 cents per bunch, find the cost of the shingles to cover 
the roof. 

56. At 20 cents per square yard, what will it cost to paint the 
roof? 

57. Cover the sides with siding at $28 per M, laying it 4 
inches to the weather. A 6 inch siding will cover only a 4 inch 
width. 

58. What will it cost to cover the gable ends with siding? 

59. At 30 cents a square yard, what will it cost to paint that 
house ? 

60. If the doors cost $7.00 apiece, and the windows $7.50, find 
the cost of the doors and windows. 

61. Cover the sitting-room with ordinary flooring 1J inches 
thick, at $30 per M. Find cost. 

62. How much would it cost to put a floor in the hall ? 

63. The dining-room? 


PRACTICAL PROBLEMS. 


37 


64. Put a 2 inch hard wood floor in the kitchen at $30 per M. 
Find cost. 

65. How much would the same quality of floor for the pantry 
cost? 

66. Find the number of feet of inch lumber necessary for a 
base-board one foot high for the sitting-room. 

67. A like base-board for the bed room. 

68. A 1J foot base-board for the dining-room. 

69. A 3 foot wainscot for the hall. 

70. Put a 4 foot wainscot in the kitchen; how much lumber 
will it take if 10 % is added for matching? 

71. At 24 cents a bunch, what would it cost to lath the kitchen? 
Do not forget the ceiling, and subtract the wainscot. 

72. At 18 cents a square yard, what would it cost to plaster the 
sitting-room? 

73. Find what it would cost to lath the kitchen, without the 
ceiling. 

74. The ceiling is covered with fine matched inch lumber cost¬ 
ing $28 per M. Add 15 % for matching and find cost. 

75. How much lumber in a 4 foot wainscot in the pantry, 
adding 10 % for matching? 

76. At 25 cents per double roll, find cost of papering the sit¬ 
ting-room. 

77. A border at 3 cents a yard is put around the top next to 
ceiling. Find cost. 

78. What would it cost to paper the dining-room if the paper 
costs 20 cents per double roll ? 

79. Cost of the border at 2 \ cents per yard. 

80. The sitting-room is carpeted with a carpet f yard wide. 
At $1.25 per yard,“find cost. 

81. Carpet paper 1 yard wide is placed under the carpet in the 
sitting-room, and costs 5 cents a yard. Find cost. 

82. The hall is covered with oil cloth 32 inches wide, and costs 
40 cents a yard. Find cost. 


38 


MENSURATION. 


83. Find the cost of the carpet paper in the hall. 

84. A carpet 1 yard wide, costing 50 cents a yard, is placed in 
the bed room. Find cost. 

85. What is the diagonal distance across the sitting-room? 

86. How far is it from the upper north-west corner to the 
lower south-east corner of the sitting-room ? 

87. Suppose a spider travels from the upper north-east corner 
of the dining-room straight down to the floor, diagonally across 
to the opposite corner, up to the upper south-west corner and diag¬ 
onally across to the starting point. What distance has it traveled ? 

88. What kind of a figure has it described ? 

89. What is its area? 

90. What is the length of its diagonal? 

91. What is the diagonal distance across a window? 

92. If the dimensions of the sitting-room were doubled, what 
would its area be ? 

93. What would be the dimensions of the bed room if it were 
twice as large ? 

94. How long must a ladder be to reach to the ceiling and 
stand 4 feet from the wall ? 

95. Find how far it is from the top of the north-east corner oi 
the 6 foot board fence to the north-east corner of the eaves. 

96. A fly travels from the lower north-east corner of the sit¬ 
ting-room up 10 feet and from there flies to the upper south-west 
corner, then down to the lower corner and diagonally across tic 
the starting point. How far has it traveled ? 

97. What kind of a figure has it described? 

98. Find its area. 

99. How much of the north end of the hall must be cut off tc 
make just a square yard? 

100. A lightning-rod is put on the house, which stands 5 feel 
on the ridge pole and is placed 6 feet into the ground. How 
long is it ? 


39 


LESSON XIV. 

CIRCLES. 



way to draw one is to inscribe it in a circle, the radius of the circle 
Deing exactly the length of one side of the hexagon. Draw one 
i little larger than this one on a piece of pasteboard. 

2. Find the area of this hexagon if the line n x is 5 and one 
>ide 6. The best way to do this is to draw the diagonals. How 
nany triangles have you now? Are they equal? What is the 
trea of each ? 


3. Cut the hexagon that you have drawn into six equal tri- 
tngles and lay them in a row thus: 



4. What is the length of a b ? What is the altitude of each 
riangle ? What is the area of that figure ? 






40 


MENSURATION. 


5. Can you see that the line a b corresponds to a c e i d b, or 
the distance around the hexagon, and the altitude corresponds 
to nx? Then, how may we find the area of any hexagon? 

Note: The right name for the line n x is apothem, and for the sum 
of the lines around it, perimeter; but since our purpose is to work towards 
a circle, we shall call n x the radius and a c e id b the circumference. 

Area of Hex. — J radius x circumference. 

6. If the circumference is 25 and the radius is 4, what is the 
area? 

7. Draw a regular figure with 12 equal sides. Can you find its 
area the same way as you did that of the hexagon ? 

8. Imagine a figure with 100 equal sides. How would you 
find the area of that ? That figure would be almost a circle. Find 
its area if the circumference is 50 and the radius 8. 

9. A circle is really nothing more than a regular figure with 
an infinite number of sides. If the area of a regular figure is found 
by multiplying \ the radius by the circumference, how would you 
find the area of a circle? 

Area of Circle = |r x Circum. 

10. If the circumference of a circle is 100 and the radius 16, 
what is the area ? 



CIRCLES, 


41 


11. The line a b is called the diameter. How many radii in a 
diameter ? Diameter — 2 r. Remember that. 

12. With a tape measure find the distance around a fruit can. 
Find the diameter. 

13. Measure the circumference of other circular objects and 
also other diameters. Place the circumferences in one column, 
and the corresponding diameters opposite each in another column. 
About how much larger is each circumference than its correspond¬ 
ing diameter ? 

14. More exactly, the circumference is about 3.1416 times the 
diameter. This number, 3.1416, is called pi and is written ir. 

(a) Circumference — nx Diameter , but 

(b) Diameter = 2 r, hence, 

(c) Circumference = tt x 2 r. 

You have learned in example nine that 

(d) Area of Circle — Circumference x -J r. 

Now, instead of using the word circumference in equation (d), 
let us substitute the value of circumference, tt x 2 r, as given in 
equation (c), and we have 

(e) Area of Circle = irx2 rx^r. 

Multiplying, 2 x J = 1 ,rxr = r\ hence, 

(f) Area of Circle — irxr 2 . 

That is, the area of a circle is obtained by multiplying the 
square of the radius by 3.1416. If the radius is 2, what is the 
area? If it is 5? If it is 10? 

15. If the diameter of a circle is 5, what is the circumference? 

16. If the radius is 16, what is the circumference? 

17. The circumference is 31.416, what is. the radius? 

18. Circumference 25.1328, find radius. 

19. Diameter 36, find radius. 

20. Radius 8, find area. 

21. Diameter 50, find area. 

22. Radius 16, find circumference. 

23. Circumference 37.6996, find radius. 

24. Find area. 

25. Circumference 62.832, find area. 

26. Radius 9, find area. 


42 


MENSURATION. 


LESSON XV. 

PROBLEMS. 

1. Study this out and see if you can understand it: 

12 = 3 x 4, then m — ax c, then A — ttxt 2 , then 

4 = 12 4- 3; c — m a; r 2 = A -r- tt. 

If the square of the radius equals the area 4- tt , then the radius 
equals the square root of the area after it is 4- i r. 

r 2 = A- f- 7 T, then 27. If A is 12.5664, find r. 

r = V (A 4- «■). ^ = 12.5664; tt = 3.1416; 

12.5664 4- 3.1416 = 4;\/4 = 2. 
Hence the radius is 2 if the area is 12.5664. 

2. What is the radius of a circle whose area is 153.9384? 

3. What is its diameter ? Its circumference ? 

4. Area is 1320.2543, find diameter. 

5. Area is 201.062, find circumference. 

6. Find the circumference of a wheel whose diameter is 5 feet. 

7. At 90 cents a rod, what will it cost to fence a circular piece 
of land whose diameter is 25 rods? What is its area? 

8. A circular park is 65 rods in diameter, how many acres 
does it contain? What is its circumference? 

9. A horse is attached to a post by a rope 20 feet long. What 
is the circumference of the circle in which it may graze ? 

10. What is its area? 

11. How many square yards are there in a circle whose diam¬ 
eter is 12 feet? 

12. Draw a square containing 81 square inches; inscribe a 
circle in this square. What is the area of this circle ? 

13. How much greater or less, at $1.20 a rod, is the cost of 
fencing a circular piece of ground whose diameter is 12 rods than 
a square field whose length is 12 rods ? 

14. One side of the square is 5; what is its area? 

15. Find the length of the diagonal of that square. Disregard 
the fraction. 

16. What is the area of the circumscribed, or largest circle? 


problems. 


43 



17. What is its circumference? 

18. Area of A? Length of the arc of A? 

19. What is the area of the inscribed, or second largest circle? 

20. What is its circumference ? 

21. Area of B? The length of its arc? 

22. Area of G? 

23. If the angle in C is 45 degrees, what is its area? 

24. The length of its arc ? 

25. Find the area of E. The circumference. 

26. Area of D. 

27. Area of F. The length of its arc. 







44 


MENSURATION. 


LESSON XVI. 

MISCELLANEOUS PROBLEMS. 

1. Find the length and width of a square box which shall con¬ 
tain as much as 2 boxes, one 2 feet and the other 2| feet square, 
the three boxes being the same height. 

2. From a lot of land 60 rods square 60 square rods were sold; 
what was the value of the remainder at $160 per acre? 

3. If 2-inch pickets are placed 2 inches apart, how many pick¬ 
ets will be required for a fence half a mile long? 

4. 30 feet from a tree 38 feet high, a pole 8 feet long is erected. 
What is the distance from the top of the pole to the top of the 
tree? 

5. How much less will the fencing of 200 acres cost in the 
square form than in the form of a rectangle whose length is 5 times 
its breadth, the price being $2.40 per rod? 

6. Find the diameter of a circular island containing 2 square 
miles. 

7. How many persons can stand in a park 20 rods long and 
8 rods wide, allowing each person to occupy a space of 3 square 
feet? 

8. How many yards of cotton 1J yards wide will it take to 
line 15 yards of cloth f of a yard wide? 

9. Show that a garden 40 feet square contains the same num¬ 
ber of square feet as a garden 20 x 80 feet. At 30 cents a foot, how 
much more would it cost to fence the second than the first ? 

10. What is the diameter of the largest circular saw that can 
be taken through a doorway 8^ feet high and 6§ feet wide? 

11. On a level playground there is a rope, 11J feet long, 
fastened to a ring at the top of a pole 9 feet high. How far from 
the foot of the pole will the rope reach to the ground ? 

12. How many square yards of plastering are there in a room 
24 feet long, 18 feet wide and 10 feet high, if the doors and win¬ 
dows take up 150 square feet? 


MISCELLANEOUS PROBLEMS. 


45 


13. A rectangular field contains 20 acres and is 50 rods wide; 
how long is it ? 

14. What must be the length of a farm 80 rods wide contain¬ 
ing 75 acres? 

15. A goat is fastened to the top of a post 40 feet high by a 
rope 50 feet long. Find the area of the greatest circle over which 
he can graze. 

16. How much larger is a square circumscribing a circle 36 
rods in diameter than a square inscribed in the same circle ? 

17. How much more will it cost, at $1£ a rod to fence a field 
in the form of a rectangle 108 rods long and 48 rods wide, than 
to fence a field of equal area in the form of a square ? 

18. The distance around a rectangle is 400 feet. The differ¬ 
ence between the length and breadth is 40 feet. Find area. 

19. How many square yards are there in the floor, walls and 
ceiling of a room 20 feet long, 15 feet wide and 10 feet high? 

20. A horse is to be tethered in the center of a rectangular lot 
240 feet long by 238 feet wide. How long must a rope be which 
will allow him to graze to the corners of the lot? 

21. If the diameter of an iron column is 3.5 inches, find the 
circumference. 

22. Find the area of the bottom of the pail in your school. 

23. A man bought a piece of land 1000 feet long and 400 feet 
wide at 20 cents a square foot; how much did it cost him? 

24. In order to make the land available for house lots, he put 
in two streets, each 50 feet wide, through the center, one in the 
direction of the length, and the other in the direction of the width. 
How many square feet did these streets occupy? 

25. Illustrate this by diagram. 

26. Flow much did the man make on the investment if, after 
spending $5000 in the construction of the streets, he sold the 
remainder at 30 cents a square foot? 

27. How many bricks, each 9 x 4J inches, will be required to 
pave a floor 36 x 27J feet? 


46 


MENSURATION. 


28. A tree 65 feet high casts a shadow 45 feet long. How 
far is it from the end of the shadow to the top of the tree ? 

29. The radius of a circle is 5 feet; find the diameter of another 
circle containing 4 times the area of the first. 

30. How many acres in a semi-circular farm, whose radius is 
50 rods? 

31. Find the area of the lid of the stove in school. 

32. A room is 48 feet long, 36 feet wide and 11 feet high. 
What is the length of a line drawn from one corner of the floor 
diagonally across to the opposite corner of the ceiling? 

33. What is the length of the diagonal of a rectangle, con¬ 
taining 7J acres, one side of the rectangle being 30 rods ? 

34. If a man walks f of a mile due north, then 1£ miles due 
west, then in a straight line to the point of starting, around how 
many acres has he walked? 

35. How many square feet are there in a gravel walk 4 feet 
wide, which runs around the outside of a garden 40 x 28 feet ? 

36. Illustrate this by diagram. 

37. The equatorial diameter of the earth is 7925 miles, how 
long is the equator ? 

38. The distance from the center of the hub of a wheel to the 
outer edge of the felly is 15 inches. How long must the tire be? 

39. How wide is a piece of land 80 rods long, which contains 
10 acres. 

40. How many feet of flooring in your school ? 

41. Find what it will cost to sod a triangular plot of ground 
whose base is 100 feet and altitude 33 feet, at 10 cents per square 
yard. 

42. How many acres in a field in the form of a trapezium, di¬ 
vided by a diagonal 130 rods long, into two triangles whose alti¬ 
tudes are 60 and 80 rods respectively ? 


MISCELLANEOUS PROBLEMS. 


47 


43. From a lot 40 rods square I sold 40 square rods. Find the 
value of the remainder at $120 per acre. 

44. If the longest side of a right triangular lot is 50 yards and 
the shortest side is 90 feet, find the length of the third side. 

45. If the circumference of a circular pond is 3T4.16 rods, 
what part of a mile must I row to pass from shore to shore through 
the center of the pond? 

46. If a horse is tethered to the middle post of a fence, from 
which he can graze out into the field in a curved line 78.539314 
feet long, how long is the tether ? 

47. The area of a rectangle equals 2704 square feet. What is 
the length of a square of equal area ? 

48. Find the number of feet of lumber in a plank 16 feet long, 
14 inches wide and 2£ inches thick. 

49. Find the value of 1£ inch plank, at $24 per M, that will 
be needed to make a land-roller whose circumference is 15 feet 
and length 12 feet. 

50. What is the area of both ends of this roller ? 

51. Find the distance around a right triangle whose altitude 
is 28 yards and base 112 feet. 

52. What will be the circumference of the largest circle that 
can be drawn on a sheet of paper 12 x 18 inches? 

53. What is the area of your school yard? 

54. How many fence posts are needed around it, if 12 feet 
apart ? 

55. How many feet of fence boards, if built 5 feet high? 

56. Build a sidewalk 4 feet wide close around your school 
aouse. How many feet of 1J inch lumber will it take ? 

57. How many shingles in the roof of your school? 

58. A square yard is cut from a board 18 feet long and 10 
nches wide. Find the length of the remainder. 

59. How many 8 inch furrows must be plowed across a square 
L0 acre field to make an acre? 


48 


MENSURATION. 


LESSON XVII. 


LAND MEASURE. 


A 

B 

D 

t 

F 

G 

H 

1 

J 

L, 

K 

C 

M 

N 


1. This represents a section of land. How many acres ? 

2. How many acres in c, the S. W. J ? 

3. What quarter of the section is a + b ? (N. W. J.) 

4. What half of the N. W. J is b ? (E. i of N. W. J.) 

5. What half of the N. W. J is a? 















LAND MEASURE. 


49 


6. How many acres in a? 

7. Describe m. (N. £ of S. E. £.) Describe n . 

8. How many acres in w? Inn? 

9. What quarter of the N. E. £ is d? (N. W. £ of N. E. £.) 

10. How many acres in it? 

11. Describe l. How many acres in it? 

12. What half of the N. E. £ of N. E. £ is g? ( S. £ of N. E. 
£ of N. E. £.) 

13. Describe e. Describe/. How many acres in each? 

14. Describe k. How many acres ? 

15. What £ of the S. W. £ of the N. E. £ is /? How many 
acres ? 

16. h is the N. £ of N. W. £ of S. W. £ of N. E. £. How 
many acres ? 

17. Describe i. How many acres in it? 

18. How many acres in hi j? In g l d ? 

19. How many acres in b d g? In m k /? 

20. In b i j? In c h d? In a c n? In / g l ? 

21. How many acres in E. £ of N. W. £ ? In S. E. £ of N. E. £ ? 

22. In N. E. £ of N. E. £ of N. E. £? 

23. In N. £ of S. E. £ of N. W. £ of N. E. £? 

24. How many times & is c? 

25. How many times his ml 

26. How many rods from one comer of the section to the next 
corner ? i. e. How many rods in a mile ? 

27. How many rods on the eastern boundary of g? On the 
southern boundary of k ? 

28. In the following problems give all distances in rods and 
measure on boundary lines only. What is the distance from N. E. 
corner of c to S. W. corner? 

29. From N. E. corner of e to S. E. of g? 

30. From N. E. of j to S. E. of c? 

31. From N. E. of h to N. E. of n? 

32. From N. W. of b to S. E. of i? 


50 


MENSURATION. 


33. What is the distance around i? Around e? Around k ? 

34. The distance around l ? Around a ? Around c ? 

35. Which farm would you rather have, d h j or i j g? 

36. How far is it in a straight line from the N. E. corner of g 
to the S. W. corner of k ? How far is it by traveling on boundary 
lines? 

37. What part of the section is a? k? I? e? i? 


LESSON XVIII. 
MISCELLANEOUS PROBLEMS. 














































































MISCELLANEOUS PROBLEMS. 


51 


1. This diagram represents four sections of land. With a ruler 
md pencil carefully divide it into 40 acre squares. 

2. Find areas in acres and distances in rods. What is the 
irea of 12? 14? 9? 2? 6? 8? 

3. Without considering the width of roads, railway or river in 
:he following series of problems, determine the area of the sheep 
iarm. 

4. The area of farm C. 

5. Farm B. 

6. Waste land. 

7. Area of dairy farm. 

8. Park. 

9. Millpond. 

10. Lumber yard. (Notice that the railway cuts across a row 
)f four forties, starting at the S. W. corner of the first forty, cross- 
ng the E. boundary of that 40 a J way up from its southern 
>oundary, crossing the E. boundary of the second 40 a J way up, 
tc.) 

11. Area of swamp land. 

12. Of farm A. 

13. Of the woods. City (on both sides of the road). 

14. The distance around the hill. 

15. Area of the hill. 

16. Area of the quarter section not occupied by the hill. 

17. If the river is 10 rods wide, how much land is taken up 
•y it? 

18. How much land is taken up by the railway, 8 rods wide? 
!y road B, 4 rods wide ? 

19. What is the distance around the city limits? 

20. The proprietor of the orchard had to build a 12 foot high 
ence on the city side. Find the cost of this fence if lumber is 
forth $25 per M, and posts cost 60 cents apiece. The posts are 
et 10 feet apart, and three 2x4 stringers are used. 


52 


MENSURATION. 


21. The summit of the hill is 10 rods south of the center; what 
is the distance from the north side to the summit if the hill is 
198 feet high? 

22. What is the distance around the park? 

23. How much less is the distance around the hill than the 
distance around the quarter that it occupies? 

24. If railroad ties cost 28 cents apiece, what would the ties 
in this railway cost if laid 2J feet apart from center to center? 

25. In what direction does the city seem to be growing? 

26. What is the length of the race track? 

27. How much land is enclosed by it? 

Note: The student will readily observe that the possibilities of this 
diagram are far from being exhausted. The author has seen fit to omit 
a number of good problems, and leave it for the student to make at least 
20 more. It will be found to be a valuable exercise. Work the problems. 


LESSON XIX. 

MISCELLANEOUS PROBLEMS. 

1. Fifty acres of land are fenced off into two square fields, the 
area of one of which being four times as much as the other. At 
60 cents a rod, what is the cost of fencing both fields if they are 
not adjacent? 

2. How many feet of lumber in the following: 

24 2 x 10 joists 18 feet long. 

4 beams 20 feet long and 9 inches square. 

200 pieces of siding 12 feet long and 4 inches wide. 

3. How much more will it cost, at $1.50 a rod, to fence 10 
acres in the form of a square, than in the form of a circle ? 

4. The diagonal of a square is 8, what is the length of a side 
of the square? 


MISCELLANEOUS PROBLEMS. 


53 


5. At $1.25 a rod, what is the cost of fencing a square field 
whose area equals that of another field, whose parallel sides are 
respectively 80 rods and 120 rods, and the distance between them 
30 rods? 

6. Find the area of a rectangle 24 x 20. Find the area of 
another, half of these dimensions. 

7. A box, without a lid, is 5 feet long, 4 feet wide and 3 
feet deep, inside dimensions. How many square feet of zinc will 
it take to line the bottom and sides of this box? 

8. Find the area of a triangle whose longest side is 25 rods 
and whose altitude is 5 feet. 

9. Find the cost of the boards of a sidewalk 4 feet wide, 
around the outside of a garden 200 yards by 90 feet. The boards 
cost $22 per M, and are 14 inches thick. 

10. Find the cost of a like walk around the inside of the 
garden. 

11. If 4 x 5 rafters are used, and they are placed 30 inches 
apart, how many feet of lumber are there in the 20 foot rafters of 
a double roof 40 feet long? 

12. A man mowing, walks .4 miles an hour. It takes him 72 
minutes to mow a plot of 1056 square yards. How broad a swath 
does he mow? 

13. A triangle contains 233J square yards. The perpen¬ 
dicular distance from the apex to the base is 20 feet. Find the 
base. 

14. Draw a circle 4 inches in diameter. Draw two of its diam¬ 
eters, the one at right angles to the other. Draw the lines con¬ 
necting the points where the extremities of the diameters meet the 
circumference. What figure is inscribed in the circle ? 

15. To find its area, what dimensions must you know? 

16. What is the base? The altitude? 

17. What is the area? 

18. Draw another circle 6 inches in diameter and find the area 
of the inscribed square. 


54 


MENSURATION. 


19. Draw an inscribed square on the small end of a log sawec 
off. Suppose the diameter to be 24 inches, what is the area of th< 
square ? 

20. If this log is 16 feet long, and the slabs are cut off leav' 
ing only a square piece of timber, how much lumber does it con 
tain ? The number of feet of lumber can safely be figured in this 
way. 

21. How much lumber in a log 18 feet long 4 feet in diameter! 

22. A log 20 inches in diameter and 20 feet long? 

23. 50 inches In diameter and 16 feet long? 

24. A log 12 feet long and 8 inches in diameter ? 

25. Find the exact number of inches and feet of inch lumbei 
needed for a box with cover, 4x5x6 feet on the outside. 

26. What are its dimensions on the inside ? 

27. How much lumber in a trough made of 2 inch planks! 
It is 16 feet long, 2 feet high, the bottom is 8 inches and top 2‘ 
inches wide, outside dimensions. 

28. A garden 40 rods long is laid out with semi-circular ends 
it is 10 rods wide. Find area. 

29. Find area of a 4 foot path around the outside of it. 

30. Around the inside of it. 

31. How much is gained by crossing a 165 x 532 yard fiel< 
diagonally, instead of going around? 

32. A street 5 miles long contains 30 10-33 acres; find width 

33. The area of a trapezoid is 2J acres, and the sum of thi 
parallel sides is 40 rods. Find altitude. 

34. A rope 73 feet long reaches from the edge of a ring t< 
the top of a center pole 55 feet high. How wide is the ring? 

35. Find the circumference of the ring. 

36. Find the area of the circle. 

37. A stick of timber 3 feet square and 16 feet long is sawe< 
into inch lumber. How much lumber in it, 1-6 being lost in saw 


MISCELLANEOUS PROBLEMS. 


55 


38. Which will carry the largest amount of water, two 3-inch 
or one 4-inch tile? Consider the speed the same in both cases. 

39. Find the difference in area of a square and an equilateral 
( equal-sided ) triangle, each of whose sides is 6 feet. 

40. Two circles with a radius of 7 feet each are placed in a 
circle with a radius of 14 feet. Find the area not covered. 

41. How many pickets 3 inches wide will it take to put up a 
fence 40 rods long, 3 inches being allowed between pickets ? 

42. Suppose this fence has to run over a hill 40 feet high, how 
many more pickets will it take ? 

43. A quarter section is divided into square 10 acre fields. 
Find the number of rails at 14 to a rod necessary to fence it. 

44. Find cost of paving a hexagonal court, each side being 60 
feet, at a cost of $3 per square yard. 

45. A pole whose circumference is 9 inches and height is 30 
feet has around it a wire in the form of a spiral which goes around 
it once every foot. Find length of wire. (Hypotenuse of a 
triangle.) 


MENSURATION OF SOLIDS. 

LESSON XX. 

PRISMS AND CYLINDERS. 

Take a common crayon box, and you have a good example of a 
prism. Set it on end; its height in that position we’ll call its alti¬ 
tude, the rectangle on which it stands is its base, and the top rect¬ 
angle is its other base. You will notice that the two bases are 
equal; this is true in all prisms. As long as the two bases are 
equal, and have the same shape and are parallel, the solid is a 
prism. If the bases are squares it is a square prism; if they are 
triangles, a triangular prism; if a rectangle, rectangular prism. 
If the bases are circular the solid is a cylinder. 


56 


MENSURATION. 


1. Back to the crayon box. What is the area of its base ? 

2. What is the area of its sides ? 

3. Find the area of the entire surface of the crayon box. 

4. What is the surface of a box whose base is 4 x 6 feet and 
whose altitude is 10 feet? 

5. Take a triangular prism. The base of the triangle which 
forms this base of the prism is 6, altitude 8. Find the area of 
both bases? 

6. What dimensions do you want, to find the area of the lateral 
sides ? Cut a triangular prism from a piece of pine and measure. 

7. Find the area of the entire surface of this prism. 

8. If the bases are circles having a diameter of 4, what is the 
area of the bases of this cylinder ? 

9. Take a fruit can. Measure its diameter and find the area 
of its bases. -Now cut a piece of paper, as wide as the can is 
high, and wrap it tightly around the can once. Cut off the paper 
not needed. Now take the paper off and hold it flat. 

10. What figure have you? 

11. Find its area. 

12. To what does the base in that figure correspond in the 
cylinder ? 

13. To what does the altitude correspond? 

14. What, then, is the entire surface of the fruit can? 

15. Take another cylinder 10 inches in diameter and 20 inches 
high. Find its entire surface. 

You have learned that the circumference of the cylinder cor¬ 
responds to the base of a rectangle, and the altitude of the cylinder 
to the altitude of the rectangle. Hence the 

Curved Surface of Cylinder = Circum. x Altitude. 

In like manner the lateral surface of any prism may be found 
by multiplying the perimeter, or the distance around the base, by 
the altitude. In the preceding chapter you have learned how to 
find the area of any shaped base. 


PRISMS AND CYLINDERS. 


57 


10. Find the surface of a cylinder whose diameter is 23 and 
whose altitude is 10. 

17. Radius 20, altitude 100. 

18. Circumference 157.08 and altitude 100. 

19. .Area of the base is 31416 and altitude 1000. 

20. Find the surface of a rectangular prism whose base is 
8x9, and altitude 40. 

21. Of a triangular prism, the base of whose base is 8 and 
altitude 6, and altitude of prism 9. 

22. Of a square prism, one side being 7 and altitude 24. 

23. Take a box 4 inches long, 3 inches wide and 5 inches high. 
Place one row of inch cubes along the length inside of the box. 
How many cubes in the row? 

24. How many such rows can you place on the bottom of that 
box? 

25. How many such layers to fill the box? 

26. How many cubic inches in the box? 

1 cubic inch x 4 = 4 cubic inches; x 3 = 12 cubic inches; 
x 5 = 60 cubic inches. For all practical purposes it is enough to 
say, the area of the base times the altitude equals the solid con¬ 
tents. The area of the base in this case is 12, and that times the 
altitude equals 60. 

Contents of Prism = Area of Base x A. 

27. The base of a prism is 5 square feet, the altitude is 26 feet. 
Find the contents. 

28*. The area of the base of a triangular prism is 24 and alti¬ 
tude 30. Find the contents. 

29. Find the volume of a hexagonal prism, the area of whose 
base is 6, and w’hose altitude is 10. 

30. Find the volume of a cylinder, the area of whose base is 
12, and whose altitude is 2C. 

31. The base of a rectangular prism is 5 x 8 feet. The alti¬ 
tude is 14 feet. Find the volume. 


58 


MENSURATION. 


32. One side of the base of a triangular prism is 12, the alti¬ 
tude of the base is 10, and the altitude of the prism is 20. Find 
the volume. 

33. The radius of the base of a cylinder is 10, the altitude is 
100. Find the volume. 

34. What is the surface of this cylinder? 


LESSON XXI. 

PROBLEMS. 

1. The circumference of a cylinder is 31.416, the altitude is 
20; find the volume. 

2. Find the surface. 

3. Find the entire surface of a prism whose base is an equi¬ 
lateral triangle, the perimeter being 18 feet, and the altitude 15 
feet. 

4. What is the volume of a triangular prism whose altitude 
is 15 feet, and the length of each side of whose base is 20 feet? 

5. Find the area of its surface. 

6. What is the volume of a solid 6 feet 3 inches long, 4 feet 
6 inches wide and 8 feet high ? 

7. Its surface. 

8. Find the curved surface of a piece of stove-pipe 6 inches 
in diameter and 2 feet long. 

9. The volume of this cylinder. 

10. A hexagonal prism is 12 inches high with 2 inch sides; 
find the entire surface. 

11. Its volume. 

12. How many cubic yards of earth will be removed by digging 
a cellar 12 feet long, 10 feet wide and 2 2-5 feet deep? 

13. How many cubic feet of air will a room 14 feet long, 12 
feet wide and 10 feet high contain? 


problems. 


so 


14. Find the number of square feet in the walls, ceiling and 
floor. 

15. What will it cost to dig a cellar 16 feet long, 12 feet wide 
and 3J feet deep, at 75 cents per cubic yard? 

16. If a block of stone 5 inches long, 3 inches wide and 2 
inches high weighs 44 ounces, what will be the weight in pounds 
of a cubical block of stone whose edge is 2£ feet? 

17. What is the volume of a 6 x 8 foot rectangular prism whose 
altitude is 20 feet? 

18. Find its surface. 

19. What is the volume of a cylinder whose length is 16 feet 
and the area of whose base is 54 square feet? 

20. Find its surface. 

21. What is the volume of a cylinder whose altitude is 60 feet 
and whose diameter is 10 feet? 

22. Find its surface. 

23. If one cubic foot of water weighs 62J pounds, what is the 
weight of water sufficient to fill a hollow square prism 18 feet 
long, and whose base is 5 feet wide ? 

24. Required the solid contents of cylinder whose altitude is 
15 feet and its radius 1 foot 3 inches. 

25. Find its surface. 

26. What weight of water will a rectangular tank contain, the 
length being 8 feet, breadth 51 feet and depth 7 feet ? 

27. What is the value of a pile of tan-bark 120 feet long, 36 
feet wide and 12 feet high, at $4.50 a cord ? 

28. A sleigh upon which 4-foot wood is piled is 12 feet long. 
How high must the wood be piled to contain 1J cords ? 

29. What is the volume of a triangular prism whose length 
is 12 feet, and each side of whose base is 3 feet? 

30. Find its surface. 

31. How many gallons of water will a cylindrical boiler con¬ 
tain if 25 inches high and 12 inches in diameter? 


60 


MENSURATION. 


32. Find how much tin it takes to make such a boiler. 

33. Take an ordinary tumbler whose bottom is smaller than 
the top; fit a piece of paper around it. Can you find the area of it ? 

34. Recall the trapezoid and find the area. 

35. The circumference of the bottom + the circumference of 
the top -f- 2 = ? 

36. Find the entire surface of tumbler. 

37. What is the entire surface of a pail the diameter of whose 
bottom is 12 inches, and top 20 inches, the slant height is 20 
inches ? 

38. How much tin in a pail the area of whose base is 78.54 
square inches, the cover 113.0976 square inches, the slant height 
is 20 inches ? 

39. Find the area of an inch rim on the cover. 

40. Find how much water the pail will hold, if altitude is 19 
inches. 

41. How much tin in the pail at your school? 

42. Find how much water it will hold. Prove by measuring. 

43. How much sheet iron in a stove-pipe 6 inches in diameter 
and 14 feet long? 

44. Find the contents. 

45. How much sheet iron does it take for the stove-pipe in 
your school? 

46. Find its contents. 

LESSON XXII. 

PYRAMIDS AND CONES. 

1. From a large potato cut a pyramid with a square base. 

2. Look at it a while and tell how to find the area of its base 
and sides. 

3. Find the surface of the ones you have. 

4. Find the surface of a pyramid whose base is 5 x 6 inches 
and slant height is 10 inches. 


PYRAMIDS AND CONES. 


61 


5. Find the surface of a triangular pyramid the altitude of 
whose base is 6, one side 12, and the slant height is 20. 

6. Cut an equilateral triangular prism from a large potato. 

7. Can you cut that into exactly three pyramids with the same 
base and altitude as that of the prism ? It can be done. 

8. Then, if a triangular prism can be divided into three equal 
pyramids having the same base and altitude that the prism has, 
the pyramid is what part of this prism ? 

Volume of Prism = B.xA., then, 

Volume of Pyramid = £ B.x A. 

9. What is the volume of a pyramid whose base is 24 and alti¬ 
tude 60? 

10. Base 86 and altitude 288? 

11. What is the volume of a cone whose base is 8 and alti¬ 
tude 27? (You have by this time learned that a cone is nothing 
more or less than a pyramid with a infinite number of sides.) 

12. Find the volume of a cone whose base is 22 and altitude 

66 . 

13. Base 7 feet and altitude 22 inches. 

14. Find the volume of a cone whose base diameter is 6 and 
whose slant height is 5. Distinguish carefully between slant 
height and altitude. The slant height is the distance from the cir¬ 
cumference of the base to the apex, and the altitude is the per¬ 
pendicular distance from the base to the apex. 

15. Take a cone. Wrap a piece of paper around its curved 
surface so that it fits exactly. 

16. Take the paper off and tell what figure you have. 

17. It is part of a circle, and can be divided into an infinite 
number of triangles. How would you find the area of it ? 

18. Find the area of the figure you have. 

19. Find the surface of a cone the circumference of whose base 
is 16 and slant height 24. 

20. Radius 24 and slant height 26. 


62 


MENSURATION. 


21. Find the surface of the great pyramid 764 feet square, and 
having a slant height of 451 feet. 

22. Find its volume. 

23. How many cords of stone are there in a pile 24 feet long, 
16 feet wide, and 3 feet high? (In estimating the amount of stone 
for a wall one cord makes 100 cubic feet of wall, and no smaller 
part than a quarter cord is allowed; it is customary to measure 
around the outside of the wall in estimating.) 

24. How many cubic feet of wall can be laid with 7J cords? 

25. How many cords of stone will build a wall 200 feet long, 
12 feet high and 3 feet thick? 

26. How many cords of stone are required for a cellar 26 x 16 
feet and 9 feet high, if the walls are 2 feet thick? 

27. Build a 6 foot high and 2 foot thick foundation under the 
second plan in this book, and estimate the amount of stone. 

28. How many cords of stone in the foundation of your school ? 


LESSON XXIII. 

SPHERES. 

1. The volume of a pyramid = ? 

2. If the base of a pyramid is 6 and the altitude 10, what is 
its volume ? 

3. If the sum of the bases of a number of pyramids is 30 and 
the altitude 10, what is the volume of all? 

4. If you can have, or can make, a number of pyramids of 
equal base and altitude, the bases as small as possible, and put 
them together with their apexes at one point, the solid you have 
resembles what? 

5. Suppose that the sum of the bases of all these pyramids 
that you put together is 1256 and the altitude of each is 10, what 
is the volume of the solid ? 


SPHERES. 


63 


6. By this time it must have occurred to you that a sphere is 
composed of an infinite number of pyramids; that the bases of 
the pyramids correspond to the surface of the sphere, and the alti¬ 
tude of the pyramid to the radius of the sphere. 

7. The volume of a sphere, then, equals i of the radius times 
the area of the surface. 

Vol. of Sphere = £ r x Surface. 

8. Now follows something that must be remembered. In the 
circle you had to remember that the circumference equals 3.1416 
times the diameter; in the sphere you must remember that the 
surface equals 3.1416 times the square of the diameter. 

Circumference of Circle = 3.1416 x D. 

Surface of Sphere = 3.1416 x D 2 . 

Learn this thoroughly, and then study the following: 

D equals 2 r; you see that. Then D squared equals (2 r) 
squared. Then, Surface of Sphere = 3.1416 x (2 r) 2 . See that? 

Above you learned that 

Volume of Sphere — J r times the Surface. 

Now, instead of writing Surface let us put into this equation 
the 3.1416 x (2 r) 2 , which equals the surface, and we have 

Vol. of Sphere = irx 3.1416 x (2 r) 2 . 

Let us make that smaller by joining the r’s. The square of 2 
is 4; the square of r is r; i times 4 is 4-thirds; r times r 2 is r 3 ; 
together, 4-thirds r 8 . Then 

Volume of Sphere = 4-thirds r 8 x 3.1416. 

9. If the radius is 12, what is the volume of the sphere? 
4-thirds x (12) 3 = 4-thirds of 1728; this x 3.1416 = ? 

10. Radius 3, what is the volume? 

11. What is the surface? 

12. Radius 6; find volume. 

13. Find surface. 

14. Diameter 24; find volume, 

15. Find surface. 


64 


MENSURATION. 


16. If volume = 4-thirds r 3 x 3.1416, then 4-thirds r* = Vol¬ 


ume ~ 3.1416, and 


Vol. -f- 3.1416 


or 


4-thirds 

Vol. ~ 3.1416 x J; or J Vol. 3.1416. 

r= V (J Vol. -r- 3.1416); or, in words, 
if the volume is given, divide f of it by 3.1416, and extract the 
cube root of it; the result is the radius. 

18. If the volujne is 113.0976, what is the radius? 

19. Find the surface. 

20. Volume is 12866.9936; find radius. 

21. Surface. 

22. Volume is 4.1888; find radius. 

23. Surface. 

24. Radius is 21, find surface. 

25. Find volume. 

26. A3 inch circle is placed within a 4 inch circle, how much 
of the larger circle is not taken up ? 

27. A circular sheet of gold 20 inches in diameter is to be 
divided equally between two children, one of whom is to receive 
his in the form of a circle, and the other in the form of a ring 
around the circle. What is the diameter of the circle and the 
width of the ring ? 

28. Find the cost of painting a church spire, at 25 cents a 
square yard, whose base is a hexagon 5 feet on each side, slant 
height 60 feet. 

29. Find the solid contents of a cone,'the diameter of whose 
base is 6 feet and its altitude 10J feet. 

30. Find the solid contents of a cone whose altitude is 24 feet, 
and the diameter of whose base is 30 inches. 

31. What is the cost of a triangular pyramid of marble, whose 
altitude is 9 feet, each side of the base being 3 feet, at $2J per cubic 
foot? 


32. Find its surface. 



65 


LESSON XXIV. 

DRILL EXAMPLES. 
Find the missing dimensions. 


RECTANGLES. 


Base. 

Altitude. 

Area. 

Diagonal. 

Distance 

Around. 

9 

12 

• • • 



., 

16 

256 



,, 

15 

... 

25 


8 

6 

... 



15 

... 

540 




24 

432 



# # 

5 

.. • 


13 

6 

9 

... 



, , 

12 

96 



9 

... 

315 



10 

... 

... 


30 

, . 

... 

144 



i 

... 

72 



9 

t . 

... 



i 

... 

3 



# # 

... 

256 



# , 

* 

... 



.05 

... 

1 



lift. 

... 

24 in. 



4 in. 

... 

... 


9 ft. 


4.2 

80 

... 

... 








66 


MENSTRATION. 



RIGHT TRIANGLES. 






Distance 

Base. 

Altitude. 

Area. 

Hypotenuse. Around. 

8 

10 

... 

... 


.. 

32 

192 

... 

... 

.. 

20 

. . . 

25 

... 

40 

30 

... 

... 

120 

,. 

24 

384 

... 

99 

9 

8 

... 

... 

... 

40 

... 

... 

50 

... 

t 

» 

... 

... 

... 

5 

> .. 

90 

... 

... 

,. 

8 

... 

10 

36 

,. 

.01 

.05 

... 

... 

.. 

40 

840 

... 

... 

1st Base 

2nd Base. 

Altitude. 

Area. 


TRAPEZOIDS. 


9 

8 


12 

.. 

,. 

6 


5 

25 

12 

.. 


30 

.. 

9 

10 


.. 

90 

i 

1 


.05 

.. 


CIRCLES. 


Radius. 

Diameter. 

Circum. 

Area. 

12 

.. 




.. 

20 




.. 

.. 


25.1328 


.. 

.. 



254.4696 

i 

.. 




. . 

§ 




. . 

.. 


.31416 


. . 

.. 



3.1416 

. . 

.005 




. , 

# # 



1 



















67 


RECTANGULAR PRISMS. 



Base. 

Altitude. Dim. of Base. 

Volume. 


16 

40 

• • • • 


.. 

16 4x12 

• • • • 


.. 

7x9 

441 


1 

i 

.... 


.05 

. . .... 

.08 



.6 f xf 

— 



CYLINDERS. 


Alt. 

Base. 

Diam. Circumference. Entire Surface. Vol. 

10 


12 . . 


10 


157.08 _ 


10 

78.54 




Base. 

5 

9 


Radius. 

6 

3 


PYRAMIDS. 

Alt. Vol. Dim. of Base. Slant Height. 


7 

.. 

.. 

.. 

8 

56 

,. 

.. 

,, 

24 

.. 

.. 

.. 

., 

4x4 

10 

10 

.. 

5x5 

.. 

# # 

90 

8x8 

.. 


SPHERES. 

Circumference. Surface. Volume. 


9.4248 


. 244.4696 . 

. . 1325.3625 

1.570S . . 





























MENSURATION. 


LESSON XXV. 
MISCELLANEOUS PROBLEMS. 

HOW TO MEASURE HEIGHT OF OBJECTS. 



Make a pole about 8 feet in length, n d. At right angles to this 
and 4 feet from one end firmly fasten a lath about 3 feet long, o v. 
To the end of this lath at v join, with a bolt, a longer lath, v r, ir 
such a manner that the long lath slides up and down the pole n d 
Set this instrument any convenient level distance, say 50 feet frorr 
the object the height which is to be measured, and placing the ey< 
at v sight the long lath v r by sliding it up and down so that il 
points to the top of the object to be measured. Now hold vr ir 
place and measure the distance from m to o. Let us say that it h 










MISCELLANEOUS PROBLEMS. 


69 


2J ft. From v to the object is 54 ft. The proportion in all prob¬ 
lems of this kind is: o v:c a:: mo :x, or the height of the object. 
3 : 54:: 2J: x = 45; add to this the 4 feet, because the eye is 4 feet 
from the ground, = 49 feet. 

1. If a spire is 60 feet from the eye and o m is 3 feet, how 
high is it ? 

2. If o m is 3£ feet? 

3. Distance 80 feet and o m2\ feet, how high? 

4. Find the height of the school, the church, and the height of 
trees. 

5. The diameter of the earth is 7,920 miles. What is the 
area of its surface? 

6. On a 12 inch school globe, each square inch represents how 
many square miles of surface ? 

7. How much of the globe’s surface represents Wisconsin? 

8. If a cannon ball weighs 22 pounds, what is the weight of 
another with a diameter twice as great? 

9. A 12 inch solid iron ball is melted and run into the form of a 
3 inch cylinder. How long is the cylinder? 

10. In cutting a foot sphere from a foot cube, what fraction 
of the cube is cut away? 

11. Four pipes, each 2 inches in diameter, empty into a tank. 
What must be the diameter of a single pipe to carry away all of 
the water ? 


MENSURATION. 


70 

ANSWERS. 


Lesson IV. 

8. 270. 


Lesson V. 


16; 

32; 

20; 24; 

28; 

16. 


15; 

21; 

12. 

30; 

42; 

24; 15; 

21; 

12; 

22%; 

31% 

; 18 

!,* 33%; 

44% 

; 27. 



4. 8; 16. 


5. 

5; 10 

; 12; 

23% 


26§; 

42; 36; 4§ 


19%; 

24; 

17% 


4f; 

40; 

37% 


37%; 

12; 

74§ 


10§; 

56; 

66§ 


240. 



6. 

16 sq. 

. ft. 


7. 

160 sq. ft. 


8. 

160; 

160; 

no. 


Lesson VI. 

1. 192; 192; 192. 

2. 200; 240. 

3. 10x20; 10x16. 

4. 4; 200; 160. 

5. 810; 450; 450. 

8. 264; 528. 

9. 22. 

10. 29. 

11. 540; 720. 

12. $30. 

13. $4; 36^. 

14. $9.50. 


15. $5.04. 

16. $4.80. 

17. $77. 

18. $15.77. 

Lesson VII. 

1. Boards, $184.32. 
Shingles, $150.80. 
Planks, $32.00. 
Total, $367.12. 

2. $40.59. 

3. $18.40. 

4 . 10 ft.; 12 ft.; 12 
ft. 

5. $35.88; 14 ft. 

6 $109.22. 

7. $75.64. 

8. $159.76. 

9. 16 ft. 

Lesson VIII. 

1. $16. 

2. $35.12+. 

3. $132. 

4 . $10.34. 

6. $22.85. 

7. $8.64. 

8. $17. 

9. $47.60. 

10 . 12 . 

11. 180 ft.; 8 d. rolls; 
6 d. rolls. 

12. $1.98. 

13. $2.80. 


Lesson IX. 

1. $23.34. 

2. $43.52. 

3. $53.85. 

4. $99.60. 

5. $7.08. 

6. $46.80. 

7. $25.76; 16 ft. 

8. $6.40. 

9. $19.80. 

10. $35.20. 

11. $77. 

12. $8.50. 

13. $6.92; 18.43; 3.69 

14. $23.26. 

15. $15.02. 

16. $1.60; $3.40. 

17. $5.49; $7.04. 

18. $56. 

19. $7.33. 

20. $23.96. 

21. $3.96. 


Lesson XI. 

8. 30. 

9. 110; 400. 

13. 24. 

14. 12y a . 

15. 35%. 

16. 73%. 

17. 256. 




ANSWERS. 


71 


Lesson XII. 

3. 20. 

4. 100. 

5. 215. 

13. a 16; & 26; c 24; 
d 18; e 21; f 20; 
g 9; h 12; i 8; 
; 48; k 45; l 80; 
m 1+; n 30; o 
20; p 75; q 48; r 
120; s 12; t 111. 

14. 48. 

16. 22+. 

17. 28+. 

18. 14+. 

Lesson XIII. 

4. 20; 7. 

8 . 8 . 

9. 10. 

10. 150. 

11. $528. 

12. 64.26+ rds. 

13. 13; 120. 

14. 336 ft. 

15. 350. 

16. 160. 

17. 128 rds. 

18. 40 ft. 

19. 960 sq. ft. 

20. 1540 sq. ft. 

21. 250 sq. ft. 

22. 520 sq. ft. 

23. 128 ft. 

24. 200 ft. 

25. 70+ ft. 

26. 14 ft., nearly. 

27. 13 ft. 


28. 5 ft. 

29. 46+ft. 

30. 58+ft. 

31. 20. 

32. $27.50. 

33. 67 ft.; 76. 

34. 12 ft.; 10 or 20 ft. 

35. $50. 

36. $19. 

37. $1.90. 

38. $10. 

39. 1536. 

40. 384. 

41. 10 ft. 

42. 587. 

43. 224. 

44. $10.91. 

45. $6.16. 

46. $6.33. 

47. 16.13. 

48. 21; 42. 

49. 18 ft. 

50. 504. 

51. 88. 

52. $24.96. 

53. $4.32. 

64. $24.19. 

65. $54.90. 

66. $33.60. 

57. $69.89. 

58. $12.10. 

69. $65.67. 

60. $139. 

61. $12.96. 

62. $6.48. 

63. $7.56. 

64. $10.80. 

65. $3.60. 

66 . 68 . 


67. 44. 

68. 78. 

69. 156. 

70. 247. 

71. $5.76. 

72. $22.92. 

73. $4.08. 

74. $5.80. 

75. 141. 

76. 3. 

77. 69*. 

78. $1.60. 

79. 45*. 

80. $53.33. 

81. $1.60. 

82. $7.20. 

83. 80*. 

84. $6.67. 

85. 24 +. 

86. 26.83 +ft. 

87. 60.88 + ft 

88. Rectangle. 

89. 221.28 + ft. 

90. 24 + ft. 

91. Nearly 6 ft. 

92. 1152 sq. ft. 

93. 12x20. 

94. 12.65 ft. 

95. 15 + ft. 

96. 70 +ft. 

97. Trapezoid. 

98. 264 sq. ft. 

99. 1% ft. 

100. 42 ft. 

Lesson XIV. 

15. 15.708. 

16. 100.531. 

17. 5. 



72 

Lesson XIV.—Con. 

18. 4. 

19. 18. 

20. 201.0619. 

21. 1963.4954. 

22. 100.531. 

23. 6. 

24. 113.0973. 

25. 314.1593. 

26. 254.469. 

Lesson XV. 

2. 7. 

3. 14; 43.9823. 

4. 41. 

5. 50.2655. 

6. 15.708. 

7. $70.69; 490.8739. 

8. 20.73 +A; 

204.2035 rds. 

9. 125.6637. 

10. 1256.6371. 

11. 12.5664. 

12. 63.6173. 

13. $12.36 less. 

14. 25. 

15. 7 near enough. 

16. 38.4845. 

17. 21.9911. 

18. 3661; about 5%. 
. 19. 19.635. 

20. 15.708. 

21. 4.9087; 3.927. 

22. 1.3412. 

23. 2.4523. 

24. 1.9635. 

25. 4.9087; 7.854. 

26. 2.4543. 

27. 4.908; 5.8905. 


MENSURATION. 

Lesson XVI. 

1. 3y 3 ft. 

2. $3540. 

3. 15840. 

4 . 42% about. 

5. 585.6. 

6. About .6 mi. 

7. 14520. 

8. 9. 

9. $12. 

10 . 10 %. 

11 . 6 %. 

12. 124§. 

13. 64 rds. 

14. 150 rds. 

15. 2827.4334 sq. ft. 

16. 648. 

17. $30. 

18. 9600 sq. ft 

19. 144.44 sq. yd. 

20. 169. 

21. 10.9956. 

23. $80,000. 

24. $70,000. 

26. $14,000. 

27. 3520. 

28 79 + ft. 

29. 20. 

30. 3926.9908. 

32. 61. 

33. 50. 

34. 360 A. 

35. 608. 

37. 24899. 

38. 94.2478. 

39. 20 rds. 

40. $18%. 

42. $66.77. 

43. $1170. 


44. 40 yds. 

45. %. 

46. 25 ft. 

47. 52 ft. 

48. 46§. 

49. $6.48. 

50. 36.19+. 

51. 255 +. 

52. 37.6991. 

58. 7 1-5. 

59. 99. 

Lesson XVII. 

1. 640 A. 

2. 160 A. 

6. 80 A. 

7. S. % of S. E. %. 

8. 80 A. 

10. 40 A. 

11. S. E. % of N. E. 
%; 40 A. 

13. N. W. % of N. E. 
10 A. 

N. E. % of N. E. 
% of N. E. %; 10 
A. 

% of N. E. %; 

14. S. % of S. W. % 
of N. E. %; 20 A. 

15. N. E. %; 10 A. 

16. 5 A. 

17. S. % of N. W. % 
of S. W. % of N. 
E. %; 5 A. 

18. 20 A; 100 A. 

19. 140 A; 140 A. 

20. 95 A; 280 A; 320 
A; 70 A. 

21. 80 A; 40 A. 

22. 10 A. 



Lesson XVII.— Con. 

23. 5 A. 

24. 8. 

25. 16. 

26. 320. 

27. 40 rds.; 80 rds. 

28. 320 rds. 

29. 120 rds. 

30. 480 rds. 

31. 280 rds. 

32. 240 rds. 

33. 120 rds. 

34. 320 rds.; 480 rds; 
640 rds. 

35. dhj 

36. 200 rds.; 280 rds. 

37. 1-8; 1-32; 1-16; 

1-64; 1-128. 

Lesson XVIII. 

2. 40 A; 20 A; 30 A; 
10 A; 5 A; 5 A. 

3. 200 A. 

4. 80 A. 

5. 140 A. 

6. 120 A. 

7. 200 A. 

8. 80 A. 

9. 480 A. 

10. 145 A. 

11. 115 A. 

12. 100 A. 

13. 200 A. 

14. 502.656 rds. 

15. 125.661 A. 

16. 34.339 A. 

17. About 3314 A. 

18. Nearly 33 A.; 
about 8*4 A. 

19. 1102 rds. 


ANSWERS, 

20. 541.80. 

21. Nearly 171 rds. 

22. 678 rds. 

23. 137.344 rds. 

24. $1219.68. 

25. Southwest. 

26. 411.3274 rds. 

28. 71.4+A. 

Lesson XIX. 

1. $288. 

2. 2060. 

3. About $27. 

4. 5.65+. 

5. About $68.75. 

6 . 120 . 

7. 74 sq. ft. 

8. About 61%. 

9. $184.27. 

10. 180.04. 

11. 1133y a . 

12. 3%. 

13. 210. 

18. 18 sq. in. 

19. 288 sq. in. 

20. 384 ft. 

21. 1728 ft. 

22. 333*4. 

23. 1666§. 

24. 32. 

25. 143 ft. 8 sq. in. 

26. 3 ft. 10 x 4 ft. 10 
x 5 ft. 10. 

27. 151*4 ft. 

28. 378.54 sq. rds. 

29. 6225.4+ sq. ft. 

30. 5877.3 sq. ft. 

31. 140. 

32. 50 ft. 

33. 15^3 rds. 


73 

34. 96. 

35. 301.5929. 

36. 7238.2295. 

37. 1440 ft. 

38. 2 3-in. 

39. 20.7 sq. ft. 

40. About 308 sq. ft. 

41. 1320. 

42. Not any more. 

43. 2240. 

44. $311 5-9. 

45. 31.32 + ft. 

Lesson XX. 

15. 785.3976 sq. in. 

16. 1553.5172. 

17. 15079.6442. 

18. 19635. 

19. 691152. 

20. 1504. 

21. 264. 

22. 770. 

27. 130 eu. ft. 

28. 720. 

29. 60. 

30. 240. 

31. 560 cu. ft. 

32. 1200. 

33. .31416. 

34. 6911,5086. 

Lesson XXI. 

1. 1570.796, 

2. 785.3996, 

3. 300 sq. ft 

4. 2598 cu. ft. 

5. 1246.4 sq. ft, 

6. 225 cu. ft. 

7. 228*4 sq. in. 


74 


MENSURATION. 


Lesson XXI.—Con. 

8. 452.4 sq. in. 

9. 678.5856 cu. in. 
10 164.78 sq. in. 

11. 124.68 cu. in. 

12. 10§ cu. yd. 

13. 1680 cu. ft. 

14. 520 sq. ft. 

15. $20. 

16 2475 lbs. 

17. 960 cu. ft. 

18. 656 sq. ft. 

19. 864 cu. ft. 

20. 525.2 sq. ft. 

21. 4T12.4 cu. ft 

22. 2042.04 sq. ft. 

23. 28125 lbs. 

24. About 75 cu. ft. 

25. 127.81 sq. ft. 

26. 19250 lbs. 

27. $1822.50. 

28. 4 ft. 

29. 45 cu. ft. 

30. 123 sq. ft. 

31. 12.22 + gal. 

32. 1168.7 sq. in. 

37. 1118.4 sq. in. 

38. 882.7896 sq. in. 

39. 37.7 sq. in. 

40. 8.29 +gal. 

43. Nearly 22 sq. ft. 

44. 2.74 + cu. ft. 


Lesson XXII. 


4. 140. 

5. 238.08 +. 

9. 480. 

10. 8256. 

11. 72. 

12. 484. 

13. 616 cu. in. 

14. 37.6992. 

19. 212.428+. 

20. Nearly 3770. 

21. 1,272,824 sq. ft. 

22. 46,695,680 cu. ft. 

23. 11% cd. 

24. 750 cu. ft. 

25. 72 cd. 

26. 14 2-5 cd. 

27. 15.3 + cd. 


Lesson XXIII. 


10. 113.0976. 

11. 113.0976. 

12. 893.78 +. 

13. 452.3904. 

14. 7238.2464. 


15. 1809.5616. 

18 3 

19. 113.0976. 

20. 4. 

21. 201.0624. 

22 . 1 . 

23. 12.5664. 

24. 5541.7824. 

25. 38792.4768. 

26. 14.15+. 

27. 2.93+. 

28. $25. 

29. 96.9604 cu. ft. 

30. 5654.8664. 

31. $28%. 

32. About 43% sq. ft. 


Lesson XXV. 


1. 60 ft. 

2. 70 ft. 

3. 62 2-9 ft. 

5. 193,919,658 sq. mi. 

6. 428647. 

7. About 1-9 sq. in. 

8. 176 lbs. 

9. 128 in. 

10. .5236. 

11 . 

12. 4 in. 


QlppIfl’S 

ri/INU/lLS... 

w "p".r nM °"/lKITnnETK. 

Vice-Principal Cook Co., Ill. Normal School. 


GIFFIN’S Manual on Arithmetic are not full of written rules to be 
memorized, but are rich in illustrations , which make it possible for the 
mind of the pupil to grasp principles, and, by his own efforts, strengthen 
and develope his mental powers. 

Part I., Lines.—By the ingenious use of common lines much arithmet¬ 
ical knowledge may be taught. The author teaches fractions by lines and 
arouses much enthusiasm in so doing. 

Part II., Area. —This is a practical manual, for from it the pupil gleans 
knowledge that will be useful to him all his life. The teaching of fractions 
is continued. It, among other things, instructs the pupil as to obtaining the 
area in lathing, plastering, paper hanging, shingling, etc. It contains a 
large number of examples for practice. 

Partlll., Percentage.—The instruction needed for ordinary business 
transactions may be obtained from this manual. It does not require the 
pupils to memorize written rules. Practical instruction is also added in the 
way of business letter writing, and other forms of business usage. 

Part IY., Volumn and Bulk.—This manual is a companion to Part II. 
It is well illustrated and has many practical problems. 

The work complete has over 1500 problems, which enable the pupil to 
acquire a good knowledge of the subjects under discussion. 

WHAT IS SAID OP THEM. 

“The copy of Supplementary Arithmetic has arrived. I am delighted 
with it. It is better than a gold mine to the pupils who are led to follow 
systematically its rich leads " -Principal R. K. Row , of the Kinsgton t 
Ontario , Training School. 

“The pupil is led to observe, to construct, and to judge for himself at 
every step.”— Illinois Public School Journal. 

“Your book is a hit. It is one that teachers will buy, because it will 
help them.”— E. T. Pierce , Principal State Normal School , Los Angeles , 
California. 

Prices.—Parts I. and II. 30 cents each, Parts III. and IV., 25 cents each. 

The four parts in one volume, cloth, $1.00. 

A. FLANAGAN* CHICAGO. 




ANNOUNCEMENT. 


A 

New and 

Original 

Book 


The 

Product of 
Classroom 
Experience 


FIRST STEPS 


IN 


ENGLISH COMPOSITION 

FOR 

GRAMMAR AND HIGH SCHOOLS, 
SEMINARIES AND COLLEGES 

BY H. C. PETERSON, Ph. O. 

Head Master in the English Manual Training High School, Louisville, Ky. 

About 128 pages. Price, Postpaid* 35 Cents, 

This very valuable and decidedly unique manual for pupils and teachers is 
believed to be the welcome solution of a problem that has long troubled 
teachers in our Grammar and High Schools, Academies, Seminaries and 
Colleges. 

The general plan of the book is seen in the Table of Contents on the 
other side of this sheet. The method was born of classroom needs. It is so 
simple, so bright, so practical, and so pleasantly productive of original work in 
composition writing and in the orderly and correct presentation of thought, 
that it will be hailed with delight by pupil and teacher alike. A Professor of 
English in one of our Universities says of the manuscript: 

“// is the brightest and happiest plan I have yet seen?' 


A. FLANAGAN CO., Publishers 

Chicago, Ill • f Over} 













“The SHADOWY KINGS oi BRITAIN.’’ 

FRESH AND NEW 
INTERESTING AND PROFITABLE 


The Story of the Britons 

By HUBERT M. SKINNER. 

OYS AND GIRLS love the stories which tell of the youth of our 
race. Of all the classic legends of ancient lands, the one which is 
most related to the literature and thought of the English-speaking 
world is the story of Britain through the fifteen centuries preceed- 
ing the Saxon conquest. It is not contained in the histories of 
England, but is connected with our enduring literature and with art in its 
higher forms. Even in the nursery are some of its names familiar. 

The Giants and the Giant=killer, 

King Lear and His Daughters, 

Sabrina, Lud, Cymbeline and Imogen, 

Old King Cole and Helena, 

King Arthur and His Knights, 
and some others are everywhere well known. 

Brut, Artegal and Elidure, Porrex and Ferrex, Claudia, Bladud, 
Caractacus and Boadicea are dear to lovers of literature. Still other 
characters of the story are subjects of interesting speculation to students of 
history. 

This connected, succinct presentation of the whole narrative, as a com¬ 
panion volume to the popular children’s stories of Troy and of ^Eneas, 
rounds out the Trojan Cycle, and will be found the most profitable of all 
for the young reader, because of the light which it throws upon literature, 
art, history and the related folklore of the ancients. From the building of 
Troy Novant (London) by the escaped Trojan captives of the Greeks to the 
“passing” of King Arthur, is told the legend of the British kings, which 
was for centuries regarded as strictly historical. The young pupil will 
enjoy the stories and the quaint specimens of old literature presented with 
them, while the older reader will find the comment and criticism helpful 

in many ways. 

The book is profusely illustrated, many of the pictures being photo¬ 
graphed from notable works of art. 

HANDSOMELY BOUND 
241 PAGES 

PRICE, 75 CENTS. 



A. FLANAGAN CO. 


CHICAGO. 










VALUABLE HELPS IN GEOGRAPHY. 


WICKS & BOYERS' HOW TO TEACH AND STUDY GEOGRAPHY. 

Part I ia on South America and Europe. It consists of Brace Outlines 
for the blackboard. Suggestive key words as indicators of supplementary 
work outside of text-books. Queer queries on geographical subjects ; these 
will stir up any school. Items of interest gathered from many sources and 
of much practical benefit. 275 pages. Price, $0.50. 

Part II. NORTH AMERICA. 

This volume is an exceedingly interesting one; every State in the United 
States is treated separately as mentioned. It cannot fail to be of very great 
interest to every teacher and pupil. Mexico and Canada also receive a lib¬ 
eral space. With these books, geography can easily be made the most 
interesting study in school. 325 pages, cloth, $0.75; Parts I and II, 
prepaid, $1.00. 


MORTON'S CHALK ILLUSTRATIONS FOR THE GEOG¬ 
RAPHY CLASS. 

A manual for teachers to accompany any series of geographies. By 
Eliza H. Morton, author of Potter’s Geographies, Butler-Sheldon Geogra¬ 
phies, Geographical Spice, Lessons on the Continents, etc. The popularity 
and desirability of blackboard illustrations in the teaching of geography 
has led the author to prepare this work, devoted almost entirely to sketches. 
It is not intended as a treatise on the art of drawing, but simply as a sugges¬ 
tive handbook to aid teachers in making freehand sketches of the outlines of 
many scenes and objects of interest. Any teacher with a little effort can 
step to the blackboard and make an illustration of much interest in connec¬ 
tion with almost any lesson in geography. Price $0.60 


MORTON'S LESSONS ON THE CONTINENTS. By Eliza H. Morton. 

In this pamphlet of 77 pages each continent is treated as follows: A Method 
of Teaching Position and Area; of Teaching Coasts, Surface, Outlines, Com¬ 
parisons, Contrasts, Rivers, Vegetation, Drainage, Lakes, Climates, Animal 
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be derived from this in taking up the different continents. $0.20. 


LITTLE JOURNEYS 

To Cuba, Puerto Rico, Hawaii, the Philippines, China, Alaska, Canada, 
Mexico, Japan, and Other Countries of Special Interest. 

These little books contain 96 pages. They are issued monthly. 
INVALUABLE VOLUMES TO BRIGHTEN GEOGRAPHY WORK. 
The readers take the trips along with the author. The books are 
handsomely illustrated, printed on good paper, in large, clear type. They 
are admirable for supplementary reading, for brightening the work in geog¬ 
raphy and history, and for home library. THE TEACHERS’ EDITION 
contains hints for Friday Afternoons Abroad , and other work. Price: 
Teachers’ edition per year, $1.50; per number, 15c. 


B. FLflNBGBN GO., Publishers, 266-268 Wabash avenue, CHICAGO, ILLS 







A LIDRART - TRAVEL 

Cloth Bound Volumes—Little Journeys to Every Land 

by Marian M. Georgk. 


Do not fail to include These BooKs in your order for 
Library BooKs. These make Handsome, Valuable BooKs 
for every Library. 

The following may be had bound in cloth covers: 

Cuba and Puerto] Rico, 

160 pages, 7^x5^ inches, cloth. 60 illustrations, hand¬ 
some binding. 50 cents. 

Hawaii and the Philippines. 

176 pages, 7^x5^ inches, cloth. Crater of Kilauea in 
colors. 78 illustrations. 50 cents. 


China and Japan. 

178 pages. 74 illustrations. Flags in colors. 50 cents. 

Mexico and Central America. 

160 pages. Colored Maps. 65 illustrations. 50 cents. 

AlasKa and Canada. 

174 pages. Colored Maps. 74 illustrations. 50 cents. 

- - 160 pages, colored map 

England and Wales, and illustrated, so as. 

Ireland and Scotland. 

162 pages. Flags in colors. 74 illustrations. 50 cents. 

France and Switzerland. 

180 pages. Flags in colors. Maps. 52 illustrations. 50 cents. 

Italy, Spain and Portugal. 

170 pages. Flags, Maps. 50 illustrations. 50 cents. 

Holland, Belgium and Denmark. 

180 pages. Flags in colors. 45 illustrations. 50 cents. 

These books have been selected for use by State Library Boards 
and Reading Circles in Ohio, West Virginia, Iowa, Missouri, Kansas, 
Nebraska, South Dakota, Minnesota and Wisconsin. 

Other countries of Burope are iucourseof preparatlonand these will 
be followed by journeys to A»sia, Africa and South America. 


A FLdNdQflN COnPdNY 

PUBLISHERS 266 W/1B/ISH /HIE. CHIC^QO 











S6e School Year BooKs 


THE FIRST 
SCHOOL TEAR 

nTV 

I I I 1 

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A UNIQUE and helpful series of 
books for teachers everywhere. 
Each volume presents a year’s 
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with ample lesson material, well selected, 
carefully planned and suitably correlated. 
Prepared by teachers of the Southwestern 
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sylvania. 

They are not a mere course of study 
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FOR THE GRADES 

A separate book is devoted to the work of each grade from the first to the 
eighth. The books for the first, second and third school years have already 
appeared. The others are in preparation, 

FIRST SCHOOL YEAR 

By Anna B. Thomas, Primary Training- Teacher. Mailing price, 60 cents. 
Gives for September—Nature Study, Fruits, Flowers, Leaves, Literature and 
History, Stones, Palms, Number Work, The Arts, Drawing, Writing, Construction 
Work, etc. Each month, September to June, on the same general plan. Cloth, 176 
pages. Price, 60 cents. 

SECOND SCHOOL YEAR 

By Henrietta M. Lilley, Training Teacher of Second Grade. (Ready.) Mail¬ 
ing price, 60 cents. Gives for September—Nature Study, The Butterfly, The Ant, 
Trees, Fruit, Poems, Literature, etc. About Nature for September, Number Work, 
Language, The Arts, Writing, Modeling, etc. Each month is treated in same way. 
Cloth, 224 pages. Price, 60 cents. 

THIRD SCHOOL YEAR 

By Ellen Reiff, Training Teacher of Third Grade. (Ready.) Mailing price, 
60 cents. On same general plan as for first and second years, but includes, in addi¬ 
tion to topics treated in former books, Geography. This book is 208 pages. Price, 
60 cents. 

The Three BooKs for Only $1.50. 


From Superintendent H. L. Lunt, Riverside, Cal. 

“I have received the First, Second and Third School Year Books, and I want 


to tell you how much I a 
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A. Flanagan Company 



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